need help with all 3 questions. thanksA Norman window has the shape of a rectangle with a semicircle on top. If the perimeter of the window (in total) is 30 feet, find the exact dimensions of the rectangular part of the window so that the
Answers
The exact dimensions of the rectangular part of the window is given by the relation 8.4 x 4.2 feet
Given data ,
A Norman window has the shape of a rectangle with a semicircle on top.
The total perimeter of the figure of window is P = 30 feet
Let "a" be the half of the width of the rectangle (radius of the semicircle) and "h" be the height of the rectangle.
Now , perimeter of window P is
2a + 2h + ( 1/2 )2πa = 2h + a ( 2 + π )
Now , the value of P = 30 feet
So , 2h + a ( 2 + π ) = 30
On simplifying the equation , we get
2h = 30 - a ( 2 + π )
h = ( 30 - a ( 2 + π )) / 2
h = 15 - ( 1 + π/2 )a be equation (1)
Now , let us find the area of the window , where A is the total area
And , A = 2ah + ( 1/2 )πa² be equation (2)
Substituting the value of h in equation (2) , we get
A = 2a ( 15 - ( 1 + π/2 )a ) + ( 1/2 )πa²
On simplifying , we get
A = 30 - ( 2 + π/2 )a²
To find the maximum area , we need to differentiate A
So , A' = 30 - ( 4 + π )a be equation (3)
Now , to find the dimensions of the window ,
The domain conditions are a > 0 , h > 0 and A > 0
A ( 30/( 4 + π ) ) = A ( 4.2 ) ≈ 63 feet²
And , the area of the window is maximum when a = 30/( 4 + π )
Substituting the value of a in equation (1) , we get
h = 15 - ( 1 + π/2 )a
h = 15 - ( 1 + π/2 ) ( ( 4 + π ) / 30 )
h = 2 ( 30/( 4 + π ) )
h = 8.4 feet
And , the radius of semi-circular part is 4.2 feet and height is 8.4 feet
Hence , the dimensions of the window is 8.4 x 4.2 feet
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The complete question is attached below :
A Norman window has a shape of a rectangle surmounted by a semicircle. (Thus the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 30 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.
Related Questions
In school of 30 students,10% are boys. How money are boys
Answers
If in the school consisting of 30 students, 10% are boys , then the number of boys in the school are 3.
The "Percent" is defined as a unit of measurement which expresses a proportion or ratio as a fraction of 100. It is commonly used to represent relative quantities or comparisons.
In a school with 30 students, if 10% of them are boys, we can calculate the number of boys by finding 10% of 30.
The 10% can be written as a decimal by dividing it by 100,
So, 10% is equivalent to 0.10.
Multiplying 0.10 by 30,
We get,
⇒ 0.10 × 30 = 3,
Therefore, the number of boys are 3.
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what is integral of 1/ (x times square root of (x^2-a^2
Answers
The integral of 1/ (x √(x²-a²)) is (1/2) ln[(x/ a) + √((x/a)² - 1)] + (1/2) sin⁻¹(x/a) + C, where C is a constant of integration.
To find the integral of 1/ (x √(x²-a²)), we can use a trigonometric substitution.
First, let's rewrite the denominator as:
√(x² - av) = a sin(θ)
where θ is an angle in the right triangle formed by a, x, and √(x² - a²).
Differentiating both sides with respect to x, we get:
(x / √(x² - a²)) dx = a cos(θ) dθ
Solving for dx, we get:
dx = (a cos(θ) / √(x² - a²)) dθ
Substituting this into our integral, we get:
∫ [1 / (x √(x²-a²))] dx = ∫ [1 / (a² sin(θ) cos(θ))] (a cos(θ) / √(x² - a²)) dθ
Simplifying, we get:
∫ [1 / (x √(x²-a²))] dx = ∫ [1 / (a sin(θ) cos(θ))] dθ
We can use the trigonometric identity:
1 / (sin(θ) cos(θ)) = 1 / (2 sin(θ) cos(θ)) + 1 / 2
to rewrite the integral as:
∫ [1 / (x √(x²-a²))] dx = (1/2) ∫ [1 / (sin(θ) cos(θ))] dθ + (1/2) ∫ dθ
Using the substitution u = sin(θ), we get:
∫ [1 / (x √(x²-a²))] dx = (1/2) ∫ [1 / (u(1-u²[tex])^{0.5}[/tex])] du + (1/2) θ + C
where C is the constant of integration.
We can solve the first integral using a substitution of v = u^2, and then use the natural logarithm to obtain:
∫ [1 / (x √(x²-a²))] dx = (1/2) ln[(u + (1-u²[tex])^{0.5}[/tex]) / u] + (1/2) θ + C
Substituting back in terms of x, we get:
∫ [1 / (x √(x²-a²))] dx = (1/2) ln[(x/ a) + √((x/a)² - 1)] + (1/2) sin⁻¹(x/a) + C
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Thirty volunteers will pick one of
7 blue, 5 green, 9 yellow, and 9 red marbles during an upcoming service project to tell them on which team they will serve. What is the probability that a volunteer is assigned to a team other than the green team?
HELP ASAP PLEASEE
Answers
Answer:
D !!
Step-by-step explanation:
what is the probability that a player wins $100 by matching exactly three of the first five and the sixth numbers or four of the first five numbers but not the sixth number?
Answers
The probability of winning $100 by matching exactly three of the first five and the sixth numbers is 0.0018. The probability of winning $100 by matching four of the first five numbers but not the sixth number is 0.0003.
To calculate the probability of winning $100 by matching exactly three of the first five and the sixth numbers, we first need to determine the total number of possible combinations for the first five numbers. Since each of the five numbers can be any number between 1 and 69, there are 69 choose 5 (written as 69C5) possible combinations, which is equal to 11,238,513. Out of these 11,238,513 possible combinations, we need to choose three numbers that will match the drawn numbers and two numbers that will not match. The probability of matching three numbers is calculated as 5C3/69C5, which is equal to 0.0018. The probability of not matching the remaining two numbers is 64C2/64C2, which is equal to 1.
Therefore, the probability of winning $100 by matching exactly three of the first five and the sixth numbers is 0.0018 x 1, which is equal to 0.0018. To calculate the probability of winning $100 by matching four of the first five numbers but not the sixth number, we need to determine the total number of possible combinations for four of the first five numbers. Since each of the four numbers can be any number between 1 and 69, there are 69 choose 4 (written as 69C4) possible combinations, which is equal to 4,782,487.
Out of these 4,782,487 possible combinations, we need to choose four numbers that will match with the drawn numbers and one number that will not match. The probability of matching four numbers is calculated as 5C4/69C4, which is equal to 0.0003. The probability of not matching the remaining number is 64/64, which is equal to 1. Therefore, the probability of winning $100 by matching four of the first five numbers but not the sixth number is 0.0003 x 1, which is equal to 0.0003.
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4x^2-100/2x^2-7x-15
I need the hole, vertical asymptote, x and y intercepts and horizontal asymptote
Answers
The hole is at (-5/2, 0), the vertical asymptotes at x = -3/2 and x = 5, the x-intercepts are (-5, 0) and (5/2, 0) and y-intercept is (0, -20/3), and the horizontal asymptote will be y = 2.
To find the hole, vertical asymptote, x and y intercepts, and horizontal asymptote of the function;
f(x) = (4x² - 100) / (2x² - 7x - 15)
Hole; Factor the numerator and denominator to simplify the function.
f(x) = [(2x + 10)(2x - 10)] / [(2x + 3)(x - 5)]
The function has a hole at x = -5/2 because this value makes the denominator zero but not the numerator. To find the y-coordinate of the hole, substitute x = -5/2 into the simplified function;
f(-5/2) = [(2(-5/2) + 10)(2(-5/2) - 10)] / [(2(-5/2) + 3)(-5/2 - 5)]
= 0
Therefore, the hole is at (-5/2, 0).
Vertical asymptotes; The function has vertical asymptotes at x = -3/2 and x = 5 because these values make the denominator zero but not the numerator.
X-intercepts; To find the x-intercepts, set the numerator equal to zero and solve for x
(2x + 10)(2x - 10) = 0
x = -5 or x = 5/2
Therefore, the x-intercepts are (-5, 0) and (5/2, 0).
Y-intercept; To find the y-intercept, set x = 0.
f(0) = (4(0)² - 100) / (2(0)² - 7(0) - 15)
= -20/3
Therefore, the y-intercept is (0, -20/3).
Horizontal asymptote; To find the horizontal asymptote, divide the leading term of the numerator by the leading term of the denominator.
f(x) ≈ 4x² / 2x² = 2
Therefore, the horizontal asymptote is y = 2.
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please show clear explained solution to this problem. Write inpolar coordinatesThe double integral over R of √x^2 + y^2 where R is the triangle with vertices (0,0), (3,0), and (3,3)
Answers
The double integral in polar coordinates is equal to (9π/2).
To solve the double integral of √(x² + y²) over the triangular region R with vertices (0,0), (3,0), and (3,3), we first convert the Cartesian coordinates to polar coordinates using x = rcosθ and y = rsinθ. The given integral becomes:
∬_R r dr dθ
Next, we determine the bounds for r and θ. Since R is a right triangle, the bounds for θ are from 0 to π/4. The bounds for r are from 0 to 3secθ, as it starts at the origin and goes to the hypotenuse of the triangle, which can be represented by y = x or rcosθ = rsinθ. Thus, the integral becomes:
∫(θ=0 to π/4) ∫(r=0 to 3secθ) r dr dθ
Solving the integral gives us:
∫(θ=0 to π/4) [(1/2)r²]_0^(3secθ) dθ = ∫(θ=0 to π/4) (9/2)sec²θ dθ = (9/2)[tanθ]_0^(π/4) = (9π/2).
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Research studies suggest that the likelihood a drug offender will be convicted of a drug offense within two years after treatment for drug abuse may depend on the person's educational level. The proportions of the total number of cases that fall into four education/conviction categories are shown in the table below:
Education Convicted Not convicted Total
10 or more years of education 0.10 0.30 0.40
Less than 10 years of education 0.25 0.35 0.60
Total 0.35 0.65 1.00
Suppose a single offender is randomly selected from the treatment program.
The probability that the offender has 10 years or more of education and is not convicted of a drug offense within two years after treatment for drug abuse equals:
[A] 0.10 [B] 0.30 [C] 0.75 [D] 0.40
Given that the offender has less than 10 years of education, what is the probability that the offender is not convicted of a drug offense within two years after treatment for drug abuse?
[A] 0.42 [B] 0.58 [C] 0.35 [D] 0.75
Answers
The answer is [B] 0.58.
For the first question, we look at the table and see that the probability of an offender having 10 or more years of education and not being convicted is 0.30.
Therefore, the answer is [B] 0.30. For the second question, we use conditional probability. We want to find the probability that an offender is not convicted given that they have less than 10 years of education. This can be represented as P(not convicted | less than 10 years of education). Using Bayes' theorem, we have:
P(not convicted | less than 10 years of education) = P(less than 10 years of education | not convicted) * P(not convicted) / P(less than 10 years of education)
We can find each of these probabilities from the table: P(less than 10 years of education | not convicted) = 0.35 / 0.65 = 0.5385 P(not convicted) = 0.65 P(less than 10 years of education) = 0.60
Plugging these values into the formula, we get: P(not convicted | less than 10 years of education) = 0.5385 * 0.65 / 0.60 = 0.58
Therefore, the answer is [B] 0.58.
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the table above gives the u.s. population by age and calendar year. examine the three values that are outlined in red and classify each. a) the number of 25-year olds in the year 2000 was 3.39 million. classify this point. relative minimum b) the number of 40-year olds in the year 2000 was 4.65 million. classify this point. relative maximum c) the number of 20-year olds in the year 2015 was 4.55 million. classify this point. saddle point
Answers
The answers for maximum are a) 3.39 million b) 4.65 million c) 4.55 million
The terms "saddle point", "maximum", and "population" are all related to the analysis of data in mathematics and statistics.
In the table given, there are different values for the US population by age and year. You are asked to classify three specific values that are outlined in red. Let's examine each one:
a) The number of 25-year olds in the year 2000 was 3.39 million. This point is classified as a relative minimum. A relative minimum is a point on a graph where the function is at its lowest value in a small surrounding area. In this case, the number of 25-year olds in 2000 is lower than the numbers of 25-year olds in the surrounding years.
b) The number of 40-year olds in the year 2000 was 4.65 million. This point is classified as a relative maximum. A relative maximum is a point on a graph where the function is at its highest value in a small surrounding area. In this case, the number of 40-year olds in 2000 is higher than the numbers of 40-year olds in the surrounding years.
c) The number of 20-year olds in the year 2015 was 4.55 million. This point is classified as a saddle point. A saddle point is a point on a graph where there is no relative maximum or minimum, but rather a change in the direction of the function. In this case, the number of 20-year olds in 2015 is not the highest or lowest in its surrounding area, but rather a point where the trend changes direction.
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please do these problems. i will make sure to leave good remarks!1. Evaluate (2x*y* – 3xy+5) dy. 2. Evaluate Liye*x+y*dy. 3. Evaluate ALE + ) dxdy.
Answers
Now, integrate e^(x+y) with respect to y:
= y(e^(x+y)) - e^(x+y) + C
To solve the given problems, follow these steps:
To evaluate this expression, integrate it with respect to y:
∫(2xy - 3xy + 5) dy = ∫(2xy) dy - ∫(3xy) dy + ∫(5) dy
= x∫(2y) dy - 3x∫(y) dy + ∫(5) dy
= x(y^2) - 3x(1/2y^2) + 5y + C
= xy^2 - (3/2)xy^2 + 5y + C
Evaluate ∫y(e^(x+y)) dy:
To evaluate this expression, integrate it with respect to y:
∫y(e^(x+y)) dy
Apply integration by parts: Let u = y and dv = e^(x+y) dy, then du = dy and v = e^(x+y)
Using the formula ∫u dv = uv - ∫v du:
= y(e^(x+y)) - ∫(e^(x+y)) dy
Now, integrate e^(x+y) with respect to y:
= y(e^(x+y)) - e^(x+y) + C
Evaluate ALE + ) dxdy:
Unfortunately, the given expression "ALE + )" is not a valid mathematical expression.
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Sample attrition would be reflected by the:a. average death rate of the population under study. b. inability to access identified members of a population. c. number of patients who die while participating in a study. d. number of patients who drop out of a study.
Answers
It may be necessary to use statistical methods to adjust for the missing data or to conduct additional analyses to assess the potential impact on the study findings.
Sample attrition refers to the loss of participants in a study over time, which can occur due to various reasons such as dropouts, non-response, and other factors. When participants drop out of a study, it can lead to a smaller sample size and potential biases in the study results.
Option D ("number of patients who drop out of a study") is the correct answer. Sample attrition is reflected by the number of participants who drop out of a study, which can be due to a variety of reasons such as loss to follow-up, participant withdrawal, or other factors.
It is important for researchers to carefully monitor sample attrition and attempt to minimize it to ensure that the study results are valid and reliable. If sample attrition is significant, it may be necessary to use statistical methods to adjust for the missing data or to conduct additional analyses to assess the potential impact on the study findings.
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100 points if you did it on edg can you please add the pictures
Answers
Let's use the letters OD, OE, and OF to represent the distances between points O and D, O and E, and O and F, respectively.
How to explain the circleWe know that OD = OE since O is located on the perpendicular bisector of line segment DE. Similarly, we know that OE = OF because O is located on the perpendicular bisector of line segment EF.
These two equations are combined to provide OD = OE = OF. This demonstrates that points D, E, and F are equally distant from point O.
By definition, all points on a circle are equally distant from the circle's center. Point O is the circle's center since it is equidistant from points D, E, and F.
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Lupita rides a taxi that charges a flat rate of 6.75 plus 3.20 per mile. if the taxi charges Lupita 40.03 in total for her trip, how many miles is her ride.
Enter your answer in the box as a decimal to the nearest tenth of a mile.
Answers
Answer:
10.4 miles
Step-by-step explanation:
We can model the Cost of Lupita's trip using the formula
C(m) = 3.20m + 6.75, where C is the cost in dollars and m is the number of miles she travels. We can allow C(m) to equal 40.03 and we will need to solve for m:
40.03 = 3.20m + 6.75
33.28 = 3.20m
m = 10.4
Gavin has 10 identical US coins in his pocket. The total value of the coins in cents is represented by 10 X. What does the variable X represent?
Answers
The variable X represent the value of one coin.
The unitary method is a method of solving problems by finding the value of one unit and then using it to find the value of any number of units. In this problem, we can use the unitary method to find the value of X.
We know that 10 coins have a total value of 10X cents. Therefore, the value of one coin is X cents. To find the value of 2 coins, we can use the unitary method as follows:
Value of 2 coins = 2 * X cents
Similarly, we can find the value of any number of coins using the unitary method.
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How do I find area of this shape
Answers
Perimeter = 72cm
Area = 374.1cm²
How to determine the perimeter of a given hexagon?To determine the perimeter of a regular hexagon the formula given below is used;
Perimeter of hexagon = 6a
where a = 12 cm
Perimeter = 6×12 = 72cm
Area = 3√3/2(a²)
where a = 12cm
area = 3√3/2×144
= 3√3× 72
= 374.1cm²
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click on ,begin emphasis,all,end emphasis, the questions that are statistical questions.answer options with 6 optionsa.how many books are on the shelf?b.how many microphones are on the stage?c.what is the total number of tomatoes on each plant?d.what is the distance from each classroom to the office?e.what is the average temperature at 10:00 a.m. in each city?f.what is the average number of hits by the first batter in each baseball game?
Answers
These questions are considered statistical because they involve collecting and analyzing numerical data in average.
The statistical questions among the options are:
e. What is the average temperature at 10:00 a.m. in each city?
f. What is the average number of hits by the first batter in each baseball game?
These questions are considered statistical because they involve collecting and analyzing numerical data. The average, or mean, is a statistical measure that summarizes a set of data by determining its central tendency. Therefore, questions that ask for the average or mean of a certain variable are considered statistical questions.
Here, we need to identify the statistical questions among the given options.
Statistical questions are those that can be answered by collecting data and using that data to analyze, compare, or summarize certain characteristics. Average are commonly used in statistical analysis.
From the options given, these are the statistical questions:
e. What is the average temperature at 10:00 a.m. in each city?
f. What is the average number of hits by the first batter in each baseball game?
These questions involve collecting data and calculating an average, which are characteristics of statistical questions.
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-8×f(1)-4×g(4)
-(functions)
Answers
Answer:
f(1)= -2
g(4)=6
-8× -2 -4×6=-8
Find the minimum distance from (-2,-2,0) to the surface z =√(1-2x - 2y).
Answers
The minimum distance from (-2,-2,0) to the surface z = √(1-2x-2y) is |-1/3(x+y+5)| / 6, where x and y are the coordinates of the closest point on the surface to (-2,-2,0).
To find the minimum distance from a point to a surface, we need to first find the normal vector to the surface at that point. Then, we can use the dot product to find the projection of the vector connecting the point and the surface onto the normal vector, which gives us the minimum distance.
In this problem, the surface is given by z = √(1-2x-2y). Taking partial derivatives with respect to x and y, we get the gradient vector:
grad(z) = (-1/√(1-2x-2y), -1/√(1-2x-2y), 1)
At the point (-2,-2,0), the gradient vector is
grad(-2,-2,0) = (-1/3, -1/3, 1)
Next, we find the vector connecting the point (-2,-2,0) to a general point on the surface (x,y,z):
v = (x+2, y+2, z)
Then, we find the projection of v onto the gradient vector:
proj(grad(z)) = (v · grad(z)) / ||grad(z)||^2 * grad(z)
= -(x+y+5)/6 * (-1/3, -1/3, 1)
Finally, we can calculate the minimum distance as the magnitude of the projection vector:
dist = ||proj(grad(z))||
= |-1/3(x+y+5)| / 6
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1. Find the intervals in which the function f(x) = 2/3x^3 +4x²+6x+5 is increasing/decreasing. Also, find the local maximum and local minimum if they exist.
Answers
The intervals in which the function f(x) = 2/3x^3 +4x²+6x+5 is increasing/decreasing is f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-1, ∞). A local maximum is at x = -3 with a value of f(-3) = 20. A local minimum is at x = -1 with a value of f(-1) = 3 2/3.
To find the intervals in which the function f(x) = 2/3x^3 +4x²+6x+5 is increasing or decreasing, we need to find its first derivative and determine its sign:
f'(x) = 2x^2 + 8x + 6
To find the critical points, we set f'(x) = 0 and solve for x:
2x^2 + 8x + 6 = 0
Dividing by 2, we get:
x^2 + 4x + 3 = 0
Factoring, we get:
(x + 3)(x + 1) = 0
So the critical points are x = -3 and x = -1.
From the sign chart, we can see that f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-1, ∞).
To find the local maximum and local minimum, we need to examine the concavity of the function by finding its second derivative:
f''(x) = 4x + 8
Setting f''(x) = 0, we find that the inflection point is at x = -2.
From the sign chart, we can see that f(x) is concave down on the interval (-∞, -2) and concave up on the interval (-2, ∞).
Therefore, we have:
A local maximum at x = -3
A local minimum at x = -1
The local maximum is f(-3) = 20, and the local minimum is f(-1) = 3 2/3.
In summary:
f(x) is decreasing on the interval (-∞, -3) and increasing on the interval (-1, ∞).
A local maximum is at x = -3 with a value of f(-3) = 20.
A local minimum is at x = -1 with a value of f(-1) = 3 2/3.
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An office manager needs to cover the front face of a rectangular box with a label for shipping. The vertices of the face are (–8, 4), (4, 4), (–8, –2), and (4, –2). What is the area, in square inches, of the label needed to cover the face of the box?
18 in2
36 in2
60 in2
72 in2
a
Answers
The area of rectangular box is 72 in²
What is the area of a rectangle in geometry?The area of a rectangle is the space on the border of the rectangle. It is calculated by finding the product of the length and width (width) of the rectangle and is expressed in square units.
First, use the distance formula to find the length of the sides of the rectangle, and second, use the rectangle area to find the area of the sticker you need.
The distance between (-8, 4) and (4, 4) is 12 units. The distance between (-8, 4) and (-8, -2) is 6 units. As we know, the area of a rectangle is length × width
∴ The area of a rectangle is 12 x 6 = 72 square units.
Therefore, the sticker area needed to cover the front of the box is 72 square inches.
Therefore the answer is 72in²
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Answer: B which is 72 in2
Step-by-step explanation: i took the test
a cat litter box has a width of 1 ft, a length of 2 ft, and a height of 1/3. you have a bag of cat litter containing 1 ft 3 of litter. will you be able to fit the entire bag of litter in the bag without any going over the top of the box? pls help me ^^
Answers
Answer:
No, the litter box will overflow.
Step-by-step explanation:
first you need to find the volume of the litter box.
FORMULA:
cat litter box: V=lxWxH
V= 1 x 2 x 1/3
V= 2/3
Since the bag of cat litter has more cat litter than the litter box can hold, the answer is no. 1 ft 3 is more than 2/3 ft.
IM SORRY IF THIS DOESN'T MAKE SENSE I WAS CONFUSED BY THE 1 ft 3.
An angle measures 144° more than the measure of its supplementary angle. What is the measure of each angle?
Answers
The two angles are 18 degrees and 162 degrees.
What are supplementary angles?
If the addition of the measures of two angles is 180 degrees, then they are supplementary angles.
Let x be the measure of the smaller angle in degrees.
Then the measure of the larger angle in degrees is:
x + 144
The two angles are supplementary, so their sum is 180 degrees:
x + (x + 144) = 180
Simplifying the left side:
2x + 144 = 180
Subtracting 144 from both sides:
2x = 36
Dividing both sides by 2:
x = 18
So the smaller angle measures 18 degrees, and the larger angle measures:
x + 144 = 18 + 144 = 162
Therefore, the two angles are 18 degrees and 162 degrees.
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Prove or disprove the quadrilateral is a rectangle
(70 points)
Answers
The quadrilateral with coordinates Q(-3,4), R(5,2), S(4,-1), and T(-4,1) is not a rectangle as adjacent sides are not perpendicular to each other.
Coordinates of the quadrilateral QRST are,
Q(-3,4), R(5,2), S(4,-1), and T(-4,1)
Quadrilateral QRST is a rectangle,
All angles are right angles.
Opposite sides are parallel and equal in length.
The slopes of the sides and the lengths of the sides.
Slope of QR
= (2 - 4)/(5 - (-3))
= -2/8
= -1/4
Slope of RS
= (-1 - 2)/(4 - 5)
= -3/-1
= 3
Slope of ST
= (1 - (-1))/(-4 - 4)
= 2/-8
= -1/4
Slope of TQ
= (4 - 1)/(-3 -(-4) )
= 3/1
= 3
Length of QR
=√((5 - (-3))^2 + (2 - 4)^2)
= √(64 + 4)
=√(68)
Length of RS
= √((4 - 5)^2 + (-1 - 2)^2)
= √(1 + 9)
= √(10)
Length of ST
= √((-4 - 4)^2 + (1 - (-1))^2)
= √(64 + 4)
= √(68)
Length of TQ
= √((-3 -(-4))^2 + (4 - 1)^2)
= √(1 + 9)
= √(10)
Slopes of opposite sides are equal .
This implies opposite sides are parallel to each other.
Opposite side lengths are also equal.
But product of the slopes of adjacent sides not equal to -1.
They are not perpendicular to each other.
Therefore, the quadrilateral with given coordinates is not a rectangle.
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The above question is incomplete , the complete question is:
Prove or disprove that the quadrilateral QRST with coordinates Q(-3,4), R(5,2), S(4,-1), and T(-4,1) is a rectangle.
The following table presents the heights (in inches) of a sample of college basketball players.Height: 68-71, 72-75, 76-79, 80-83, 84-87Freq: 3,5,2,2,2
Answers
The approximate variance that needs to be evaluated for the given set of heights is 18 , since 18 is close to 19 with a difference one 1 , then the correct answer is Option C.
Now, the formula for variance is
Var (X) = E [ (X – μ)² ]
Here
Var (X) = variance,
E = expected value,
X = random variable
μ = mean
Then, we need to evaluate the mean of heights and then further find the sum of squares of deviations from mean for each height value.
And divide this sum by n-1
here
n = sample size
Mean height = (2 x 69.5 + 2 x 73.5 + 4 x 77.5 + 2 x 81.5 + 3 x 85.5)/13 = 79
The addition of squares of deviations from mean is
(68-79)² + (68-79)² + (72-79)² + (72-79)² + (76-79)² + (76-79)² + (76-79)² + (76-79)² + (80-79)² + (80-79)² + (84-79)² + (84-79)² + (84-79)²
= 220
Hence,
variance = sum of squares of deviations from mean / n-1
= 220/12
= 18.33
≈ 18
The approximate variance that needs to be evaluated for the given set of heights is 18 , since 18 is close to 19 with a difference one 1 , then the correct answer is Option C.
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The complete question
The following table presents the heights (in inches) of a sample of college basketball players. Height Freq 68-71 2 72-75 2 76-79 4 80-83 2 84-87 3 Considering the data to be a population, approximate the variance of the heights
a) 5.6
b) 5.4
c) 19.2
d) 31.6
The measure of an angle is 71°. What is the measure of its complementary angle? Answer:__
This is in IXL
Answers
Answer: 19 degrees
Step-by-step explanation:
If two angles are complementary, they form a 90° angle. So the angle that is complementary to 71° is 19° because 90-71=19.
Use the law of cosines to determine the length of side b of triangle ABC, where angle B = 73.5 degrees, side a - 28.2 feet, and side c = 46.7 feet.
Answers
Using the law of cosines, the length of side b of triangle ABC is approximately 47.20 feet.
To find the length of side b of triangle ABC using the Law of Cosines, you can apply the following formula:
b² = a² + c² - 2ac * cos(B)
Given the information, angle B = 73.5 degrees, side a = 28.2 feet, and side c = 46.7 feet. Plug these values into the formula:
b² = (28.2)² + (46.7)² - 2(28.2)(46.7) * cos(73.5)
Calculate the values and solve for b:
b² ≈ 795.24 + 2180.89 - 2633.88 * 0.2840
b² ≈ 2228.07
Now, take the square root to find the length of side b:
b ≈ √2228.07
b ≈ 47.20 feet
So, the length of side b of triangle ABC is approximately 47.20 feet.
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The following table contains the probability distribution for X = the number of traffic accidents reported in a day in Corvallis, Oregon.
X 0 1 2 3 4 5
P(X) 0.10 0.20 0.45 0.15 0.05 0.05
a. What is the probability of 3 accidents?
b. What is the probability of at least 1 accident?
c. What is the expected value (mean) of the number of accidents?
d. What is the variance of the number of accidents?
e. What is the standard deviation of the number of accidents?
Answers
a. The probability of 3 accidents is P(X=3) = 0.15.
b. The probability of at least 1 accident is equal to 1 minus the probability of no accidents is 0.90.
c. The expected number of traffic accidents reported in a day in Corvallis is 1.95.
d. The variance of the number of traffic accidents reported in a day in Corvallis is 1.6525.
e. The standard deviation of the number of traffic accidents reported in a day in Corvallis is 1.284.
a. The probability of 3 accidents is P(X=3) = 0.15.
b. The probability of at least 1 accident is equal to 1 minus the probability of no accidents, which is P(X≥1) = 1 - P(X=0) = 1 - 0.10 = 0.90.
c. The expected value (mean) of the number of accidents is calculated as the sum of the products of the possible values of X and their probabilities, which is:
E(X) = 0(0.10) + 1(0.20) + 2(0.45) + 3(0.15) + 4(0.05) + 5(0.05) = 1.95.
Therefore, the expected number of traffic accidents reported in a day in Corvallis is 1.95.
d. The variance of the number of accidents is calculated as the sum of the squares of the differences between each possible value of X and the expected value, weighted by their probabilities, which is:
Var(X) = [ (0-1.95)²(0.10) + (1-1.95)²(0.20) + (2-1.95)²(0.45) + (3-1.95)²(0.15) + (4-1.95)²(0.05) + (5-1.95)²(0.05) ]
= 1.6525.
Therefore, the variance of the number of traffic accidents reported in a day in Corvallis is 1.6525.
e. The standard deviation of the number of accidents is the square root of the variance, which is:
SD(X) = √(1.6525) = 1.284.
Therefore, the standard deviation of the number of traffic accidents reported in a day in Corvallis is 1.284.
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Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.) f (x) = 2x^4 + 2x^3 - x / x^3 , x>0f(x) = ____
Answers
x² + 2 ln|x| - 1/x + C is the most general antiderivative of the function.
How to find the antiderivative of function?
To find the antiderivative of the given function, we need to find a function F(x) such that F'(x) = f(x).
We can start by separating the function into three terms f(x) = 2x⁴/x³ + 2x³/x³ - x/x³
Simplifying each term,
f(x) = 2x + 2/x - 1/x²
Now we can find the antiderivative of each term separately,
∫ 2x dx = x² + A
∫ 2/x dx = 2 ln|x| + B
∫ -1/x^2 dx = 1/x + D
Putting it all together,
∫ f(x) dx = x² + 2 ln|x| - 1/x + C
where C = A + B + C is the constant of integration.
To check our answer, we can differentiate it and see if we get back the original function (d/dx) [x² + 2 ln|x| - 1/x + C] = 2x + 2/x + 1/x²
= 2x⁴/x³ + 2x³/x³ - x/x³
= f(x)
So our antiderivative is correct.
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Determine whether the integral is convergent or divergent. 3 20 dx V3 - x O convergent O divergent f' If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)
Answers
Using the limit comparison test, it is determined that the integral ∫(3 to 20) dx / (√(3-x)) is convergent. The evaluated value is approximately 7.98.
To determine whether the integral ∫(3 to 20) dx / (√(3-x)) is convergent or divergent, we can use the limit comparison test. Let's compare it with the integral ∫(3 to 20) dx / x^(1/2):
lim x->3+ (√(x-3)) / (√x) = lim x->3+ (√(1+(x-4))) / (√x) = 1
Since the limit of the ratio is a positive finite number, and the integral ∫(3 to 20) dx / x^(1/2) is convergent (it is the integral of the p-series with p=1/2), we conclude that ∫(3 to 20) dx / (√(3-x)) is also convergent. Therefore, we need to evaluate it:
∫(3 to 20) dx / (√(3-x)) = 2(√17 - √2) ≈ 7.98
So the integral converges to 2(√17 - √2).
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Find the intervals on which the function f(x)=x^4-8x^2+16 is increasing an decreasing. Identify the function's local extreme values. If any, saying where they are taken on. Which, if any, of the extreme values are absolute?
Choose the correct answer reguardimg intervals that are increasing and decreasing
a. The function f is increasing on the subintervals (-\infty,-2],[0,2] and decreasing on the subintervals [-2,0],[2,\infty)
b the function f is increasing on the subintervals (-\infty,-2],[2,\infty)and decreasing on the subintervals [-2,0],[0,2]
c. The function f is decreasing on the subintervals (\infty,-2],[2,\infty)and increasing on the subintervals [-2,0],[0,2].
d. The function f is decreasing on the subintervals (-\infty,-2],[0,2] an increasing on the subintervals [-2,0],[2,\infty)
Choose the correct answer reguarding local extreme values
a the function f has a local minimum at x=-2 and x=2, and it has a local maximum at x=0
b. The function f has a local maximum at x=-2 and x=2, and it has a local minimum at x=0
c the function f has no local extrema.
Choose the correct answer reuaring absolute extreme values
a, the function f has no absolute extrema
b the function f has an absolute minimum at x=-2 and x=2 and no absolute maximum
c the function f has an absolute minimum at x=-2 and x=2 and an absolute maximum at x=0
d. The function f has an absolute maximum at x=-2 and x=2 and no absolute minimum
Answers
The function f is decreasing on the subintervals (∞,-2],[2,∞)and increasing on the subintervals [-2,0],[0,2]. (option c).
In this case, we are given the function f(x) = x⁴ - 8x² + 16, and we are asked to find the intervals on which it is increasing and decreasing.
To determine this, we need to take the derivative of the function f(x) and find its critical points. The critical points are the values of x where the derivative is equal to zero or undefined.
Taking the derivative of f(x), we get f'(x) = 4x³ - 16x. Setting this equal to zero, we can factor out 4x to get 4x(x² - 4) = 0. Solving for x, we get x = 0 and x = ±2 as critical points.
Next, we create a sign chart to determine the intervals of increase and decrease. We plug in test values from each interval into the derivative f'(x) and determine if it is positive or negative.
When x < -2, f'(x) is negative, so f(x) is decreasing. When -2 < x < 0, f'(x) is positive, so f(x) is increasing. When 0 < x < 2, f'(x) is negative, so f(x) is decreasing. When x > 2, f'(x) is positive, so f(x) is increasing.
Therefore, the correct answer is (d) the function f is decreasing on the subintervals (-∞,-2],[0,2] and increasing on the subintervals [-2,0],[2,∞).
To identify the local extreme values, we look at the behavior of the function around the critical points. At x = 0, we have a local minimum, and at x = ±2, we have local maximums.
We can determine if these local extreme values are absolute by looking at the behavior of the function as x approaches positive or negative infinity.
In this case, as x approaches infinity, the function f(x) approaches infinity, and as x approaches negative infinity, the function approaches positive infinity.
Hence the correct option is (c).
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Calculate the Laplace transform of the following functions. (a) f(t)=sin(2t)cos (2t) (b) f(t)=cos2 (3t) (c) f(t)=te2tsin (3t) (d) f (t)=(t+3)u7(t)
Answers
The Laplace transform of the following functions
1. f(p) = 2p/ (p² + 4)²
2. f(p) = -54/ (p² + 9)
1. f(t)=sin(2t)cos (2t)
Using Laplace Transform
sin 2t = 2/ p² + 2² = 2/ p² + 4
and, cos 2t = p/ p² + 2² = p/p² + 4
So, f(p)= 2/ p² + 4 x p/ p² + 4
f(p) = 2p/ (p² + 4)²
2. f(t)= cos² (3t)
Using Laplace Transform
cos² (3t) = (-1)² d/dt(-6 sin (3t)) = -18 cos(3t)
and, -18 cos (3t)= -18 x 3/p² + 9 = -54/ (p²+9)
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The Laplace transform of the given functions are [tex]\frac{2}{s^2+16}[/tex], [tex]\frac{3\cos \left(2\right)}{s^2}[/tex] and [tex]\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]
Given are the functions, we need to find the Laplace transformations of the function,
a) f(t) = sin(2t) cos(2t)
[tex]L\left\{\sin \left(2t\right)\cos \left(2t\right)\right\}[/tex]
Use the following identity : cos (2) sin (x) = sin (2x)1/2
[tex]=L\left\{\sin \left(2\cdot \:2t\right)\frac{1}{2}\right\}[/tex]
[tex]=L\left\{\frac{1}{2}\sin \left(4t\right)\right\}[/tex]
Use the constant multiplication property of Laplace Transform:
[tex]\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant}\:a:\quad L\left\{a\cdot f\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}[/tex]
[tex]=\frac{1}{2}L\left\{\sin \left(4t\right)\right\}[/tex]
[tex]=\frac{1}{2}\cdot \frac{4}{s^2+16}[/tex]
[tex]=\frac{2}{s^2+16}[/tex]
b) f(t) = cos2 (3t)
[tex]L\left\{\cos \left(2\right)\left(3t\right)\right\}[/tex]
Use the constant multiplication property of Laplace Transform:
[tex]\mathrm{For\:function\:}f\left(t\right)\mathrm{\:and\:constant}\:a:\quad L\left\{a\cdot f\left(t\right)\right\}=a\cdot L\left\{f\left(t\right)\right\}[/tex]
[tex]=\cos \left(2\right)\cdot \:3L\left\{t\right\}[/tex]
[tex]=\cos \left(2\right)\cdot \:3\cdot \frac{1}{s^2}[/tex]
[tex]=\frac{3\cos \left(2\right)}{s^2}[/tex]
c) f(t) = [tex]te^{2t}sin (3t)[/tex]
[tex]L\left\{e^{2t}t\sin \left(3t\right)\right\}[/tex]
Use the Laplace Transformation table:
[tex]L\left\{t^kf\left(t\right)\right\}=\left(-1\right)^k\frac{d^k}{ds^k}\left(L\left\{f\left(t\right)\right\}\right)[/tex]
[tex]\mathrm{For\:}te^{2t}\sin \left(3t\right):\quad f\left(t\right)=e^{2t}\sin \left(3t\right),\:\quad \:k=1[/tex]
[tex]=\left(-1\right)^1\frac{d}{ds}\left(L\left\{e^{2t}\sin \left(3t\right)\right\}\right)[/tex]
[tex]=\left(-1\right)^1\left(-\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}\right)[/tex]
[tex]=\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]
Hence, the Laplace transform of the given functions are [tex]\frac{2}{s^2+16}[/tex], [tex]\frac{3\cos \left(2\right)}{s^2}[/tex] and [tex]\frac{3\left(2s-4\right)}{\left(s^2-4s+13\right)^2}[/tex]
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