The length of the platform is given by the expression -2 + sqrt(3x^2 + 10x) meters.
Let's start by using the formula for the area of a rectangle, which is:
Area = length x width
We are given that the width of the platform is (x+4) meters, so we can write:
Area = length x (x+4)
We are also given that the area of the platform is 3x^2+10x−8, so we can set these two expressions equal to each other and solve for the length:
3x^2+10x−8 = length x (x+4)
Expanding the right side, we get:
3x^2+10x−8 = length x^2 + 4length
Subtracting 4length from both sides, we get:
3x^2+10x−8−4length = length x^2
Rearranging, we get a quadratic equation:
length x^2 + 4length − 3x^2 − 10x + 8 = 0
To solve for length, we can use the quadratic formula:
length = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 4, and c = -3x^2 - 10x + 8. Plugging in these values, we get:
length = (-4 ± sqrt(4^2 - 4(1)(-3x^2 - 10x + 8))) / 2(1)
Simplifying under the square root:
length = (-4 ± sqrt(16 + 12x^2 + 40x - 32)) / 2
length = (-4 ± sqrt(12x^2 + 40x)) / 2
length = (-4 ± 2sqrt(3x^2 + 10x)) / 2
length = -2 ± sqrt(3x^2 + 10x)
Since the length must be positive, we take the positive square root:
length = -2 + sqrt(3x^2 + 10x)
Therefore, the length of the platform is given by the expression -2 + sqrt(3x^2 + 10x) meters.
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Pls help i really need help on this
Where the function f(x) = x² + 2x - 3 is given, note that the x-intercepts of the function f(x) are -3 and 1, and the minimum value of the function is -4. See the attached graph.
What is the explanation for the above response?
To find the minimum and maximum points of the function f(x), we can complete the square:
f(x) = x^2 + 2x - 3
= (x + 1)^2 - 4
We can see that the function is in the vertex form f(x) = a(x - h)^2 + k, where the vertex is (-1, -4).
Since the coefficient of the x^2 term is positive, the parabola opens upwards, and the vertex is the minimum point. Therefore, the minimum value of the function f(x) is -4.
To find the x-intercepts, we can set f(x) = 0:
(x + 1)^2 - 4 = 0
(x + 1)^2 = 4
Taking the square root of both sides, we get:
x + 1 = ±2
x = -1 ± 2
Therefore, the x-intercepts of the function f(x) are x = -3 and x = 1.
In summary, the x-intercepts of the function f(x) are -3 and 1, and the minimum value of the function is -4, which occurs at the vertex (-1, -4).
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A food truck owner charges z dollars per burrito combo and $1. 50 for a side of guacamole. The expression 5 (x + 1. 50) represents the
total cost of 5 burrito combos and 5 sides of guacamole.
Which expression also represents the total cost of 5 burrito combos, which cost a dollars each, and 5 sides of guacamole, which cost $1. 50
each?
0
The expression that fits perfect for the given requirement is 5z + 7.50, under the condition that a food truck owner charges z dollars per burrito combo and $1.50 for a side of guacamole.
Here we have to apply the principles of solving algebraic equations, due to the expression provided.
From the given information, here the food truck owner charges z dollars per burrito combo along with $1.50 for a side of guacamole.
According to the information it is given that the expression is 5(z+1.50) which helps to state the total cost of 5 burrito combos and 5 sides of guacamole.
Lets now formulate the expression for the given required equation
= 5(z + 1.50)
= 5z + 7.50.
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A food truck owner charges z dollars per burrito combo and $1.50 for a side of guacamole. The expression 5(z+1.50) represents the total cost of 5 burrito combos and 5 sides of guacamole. What expression also represent the cost of 5 burrito combos and 5 sides if guacamole that cost 1.50 each.
A circle has a diameter of 4 inches. Which statement about the area and circumference of the circle is true?
O A comparison of the area and circumference of the circle is not possible because there is not enough information to
find both.
O The numerical values of the circumference and area are equal.
O The numerical value of the circumference is greater than the numerical value of the area.
The numerical value of the circumference is less than the numerical value of the area.
Answer:
The numerical values of the circumference and area are equal
Step-by-step explanation:
Circumference: 12.57
Area: 12.57
12.57=12.57
Hope this helps! :)
Find an equation in slope-intercept form for the line passing through each pair of points: (4, 7), (1, 4)
Given the measure of an acute angle in a right triangle, we can tell the ratios of the lengths of the triangle's sides relative to that acute angle.
Here are the approximate ratios for angle measures
55
°
55°55, degree,
65
°
65°65, degree, and
75
°
75°75, degree.
Angle
55
°
55°55, degree
65
°
65°65, degree
75
°
75°75, degree
adjacent leg length
hypotenuse length
hypotenuse length
adjacent leg length
start fraction, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction
0.57
0.570, point, 57
0.42
0.420, point, 42
0.26
0.260, point, 26
opposite leg length
hypotenuse length
hypotenuse length
opposite leg length
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, space, l, e, n, g, t, h, end text, end fraction
0.82
0.820, point, 82
0.91
0.910, point, 91
0.97
0.970, point, 97
opposite leg length
adjacent leg length
adjacent leg length
opposite leg length
start fraction, start text, o, p, p, o, s, i, t, e, space, l, e, g, space, l, e, n, g, t, h, end text, divided by, start text, a, d, j, a, c, e, n, t, space, l, e, g, space, l, e, n, g, t, h, end text, end fraction
1.43
1.431, point, 43
2.14
2.142, point, 14
3.73
3.733, point, 73
Use the table to approximate
m
∠
L
m∠Lm, angle, L in the triangle below.
3.2
3.2
11.9
11.9
L
L
K
K
J
J
Choose 1 answer:
The angle measure of L in the triangle is approximately 75°.
Based on the given table, we can see that the ratio of the opposite leg length to the adjacent leg length for an angle measure of 75° is approximately 3.73. Looking at the triangle in the question, we can see that the side opposite to angle L is the hypotenuse and the adjacent leg is LK.
Therefore, the ratio of the opposite leg length to the adjacent leg length for angle L is equal to the ratio of the hypotenuse length to the length of segment LK.
From the figure, we can see that the length of segment LK is approximately 3.2 units. Therefore, the length of the hypotenuse is approximately 3.73 times the length of segment LK, or:
hypotenuse length ≈ 3.73 × 3.2 ≈ 11.9
Therefore, the angle measure of L is approximately 75°.
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All the dimensions of a cube increase by a factor 3/2 how many times greater is the surface area? explain
If all the dimensions of a cube increase by a factor of 3/2, the surface area will increase by a factor of 9/2.
If all the dimensions of a cube increase by a factor of 3/2, then the new dimensions of the cube will be 3/2 times the original dimensions.
Let's say the original side length of the cube was "s". Then the new side length would be (3/2)*s.
The surface area of a cube is given by the formula 6s^2, where s is the side length.
So the original surface area of the cube would be:
6s^2
And the new surface area of the cube would be:
6(3/2s)^2
= 6(9/4)s^2
= 27/2 s^2
To find how many times greater the new surface area is compared to the original surface area, we can divide the new surface area by the original surface area:
(27/2 s^2) / (6s^2)
= (9/2)
So the new surface area is 9/2 times greater than the original surface area.
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ETA 4
Answer the following:
Myla bought an item P2,500. She decided to sell it wil 2% markup. How much will Myla’s selling price be?
Joan sold her old iphone for P5000 at 8% markdown rate. Find the markdown and the original cost of the phone.
A student assistant bought an item for P520 but later decided to sell it at P550. What is the markup?
Mother organize a garage sale and earned P120 on one item at 60% markdown. How much did mother buy the item?
The cost of a t-sirt from the manufacturer is P400. If loan wants a 30% markup based on the selling price, how much will her selling price be?
1. Myla bought an item for P2,500 and decided to sell it with a 2% markup. The selling price will be P2,500 + (2% of P2,500) = P2,500 + P50 = P2,550.
2. Joan sold her old iPhone for P5,000 at an 8% markdown rate. To find the markdown and the original cost, we first calculate the markdown: P5,000 = 92% of original price. So, the original price was P5,000 ÷ 0.92 ≈ P5,434.78. The markdown is P5,434.78 - P5,000 = P434.78.
3. The student assistant bought an item for P520 and sold it for P550. The markup is P550 - P520 = P30.
4. Mother earned P120 on an item at a 60% markdown. Let X be the original price, then X * 60% = P120. X = P120 ÷ 0.60 = P200. So, the mother bought the item for P200.
5. The cost of a t-shirt from the manufacturer is P400. If Loan wants a 30% markup based on the selling price, we'll let X be the selling price, then X - 30% of X = P400. So, 0.7X = P400. X = P400 ÷ 0.7 ≈ P571.43. Loan's selling price will be P571.43.
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Two observers at point A and B, 150 km apart, sight a balloon between them at angles of elevation 42° and 76° respectively.
How far is the observer A from the balloon? Round answer to the nearest tenth
Please show step by step
Two balloons A and B apart 150km with given angle of elevation represents observer A is at a distance of 122.5 km approximately from balloon.
Number of observers = 2
Distance between two observers A and B = 150km
Angles of elevation are 42° and 76°.
Let us consider 'h' be the height of the balloon
Let the distance from observer A to the balloon x.
Use trigonometry to find the value of x.
From observer A, the angle of elevation to the balloon is 42°.
This means that the height of the balloon above observer A is ,
h = x × tan(42°)
From observer B,
The angle of elevation to the balloon is 76°.
This means that the height of the balloon above observer B is ,
h = (150 - x) × tan(76°)
Since both expressions give the same value for h, set them equal to each other,
⇒ x × tan(42°) = (150 - x) × tan(76°)
Simplifying this equation, we get,
⇒ x × (0.9004 ) = (150 - x) × 4.0107
⇒ 0.9004x = 601.605 - 4.0107x
⇒ 4.9111x = 601.605
⇒ x ≈ 122.5 km
Therefore, the distance from observer A to the balloon as per given angle of elevation is approximately 98.3 km.
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Directed Line Segments Given the points A(-1, 2) and B(7. 8), find the coordinates of the point Pon directed line segment AB that partitions AB in the ratio 1:3.
The coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).
To find the coordinates of the point P on the directed line segment AB that partitions AB in the ratio 1:3, we can use the concept of section formula.
Let's assume the coordinates of point P are (x, y). According to the section formula, the coordinates of P can be calculated as follows:
x = (3x2 + 1x1) / (3+1) = (37 + 1(-1)) / 4 = (21 - 1) / 4 = 20/4 = 5
y = (3y2 + 1y1) / (3+1) = (38 + 12) / 4 = (24 + 2) / 4 = 26/4 = 13/2 = 6.5
Therefore, the coordinates of point P on the directed line segment AB, which divides AB in the ratio 1:3, are (5, 6.5).
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Plss someone answer this math question
The value of the reflex angle in this figure is 273 degrees
What is a reflex angle?A reflex angle is an angle that is more than 180 degrees and less than 360 degrees. For example, 270 degrees is a reflex angle. In geometry, there are different types of angles such as acute, obtuse and right angles, which are under 180 degrees.
In this given figure, there's acute angle and an obtuse angle, therefore a reflex angle must be present.
To find the reflex angle in the figure, we have to trace the green part of the figure which will give us;
180° + 93°
i.e the sum of angle on a straight line with an obtuse angle
Reflex angle = 180 + 93 = 273°
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ln(n^3 8) -ln(6n^3 13n) determine that the sequence diverges
Since ln(1/6) is a finite value, the sequence does not diverge. It converges to ln(1/6) as n approaches infinity.
To determine if the sequence diverges, we need to take the limit of the expression as n approaches infinity.
Using the logarithmic identity ln(a/b) = ln(a) - ln(b), we can simplify the expression as follows:
[tex]ln(n^3 8) - ln(6n^3 13n) = ln(n^3) + ln(8) - ln(6n^3) - ln(13n)[/tex]
= [tex]ln(n^3) - ln(6n^3) + ln(8) - ln(13n)[/tex]
= [tex]ln(n^3/6n^3) + ln(8/13n)[/tex]
=[tex]ln(1/6) + ln(8/13n)[/tex]
As n approaches infinity, ln(8/13n) approaches 0, so the limit of the expression is:
lim n→∞ [ln(1/6) + ln(8/13n)]
= ln(1/6)
Since ln(1/6) is a finite value, the sequence does not diverge. It converges to ln(1/6) as n approaches infinity.
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Heights for 16-year-old boys are normally distributed with a mean of 68. 3 in. And a standard
deviation of 2. 9 in.
Find the z-score associated with the 96th percentile.
Find the height of a 16-year-old boy in the 96th percentile.
State your answer to the nearest inch
The height of a 16-year-old boy in the 96th percentile is approximately 73 inches. Rounded to the nearest inch, the answer is 73 inches.
Find out the height of a boy in the 96th percentile?To find the z-score associated with the 96th percentile, we need to find the z-score such that the area to the right of it under the standard normal distribution is 0.96. Using a standard normal distribution table or calculator, we find that the z-score is approximately 1.75.
Next, we can use the z-score formula to find the height of a 16-year-old boy in the 96th percentile:
z = (x - μ) / σ
where z is the z-score, x is the height we want to find, μ is the mean height, and σ is the standard deviation.
Plugging in the values we have:
1.75 = (x - 68.3) / 2.9
Multiplying both sides by 2.9, we get:
x - 68.3 = 5.075
Adding 68.3 to both sides, we get:
x = 73.375
So the height of a 16-year-old boy in the 96th percentile is approximately 73 inches. Rounded to the nearest inch, the answer is 73 inches.
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The probability that Trevor studies for at least 50 minutes and passes his Algebra test is 0.88. The probability that he studies for at least 50 minutes is 0.92.
Step-by-step explanation:
If we let A be the event that Trevor studies for at least 50 minutes, and let B be the event that he passes his Algebra test, then we know:
P(A and B) = 0.88
P(A) = 0.92
We want to find the probability that Trevor passes his Algebra test given that he studied for at least 50 minutes, or in other words, we want to find P(B|A).
We can use Bayes' theorem to find this probability:
P(B|A) = P(A and B) / P(A)
Substituting in our values, we get:
P(B|A) = 0.88 / 0.92
Simplifying this fraction, we get:
P(B|A) = 0.9565
Therefore, the probability that Trevor passes his Algebra test given that he studied for at least 50 minutes is approximately 0.9565.
The area of a rectangle is 42 square meters. The length is 7 meters. What is the width?
35 meters
14 meters
15 meters
6 meters
Answer: 6
Step-by-step explanation:42/7=6
2) You buy a brand new Audi R8 for $148,700 before taxes. If the car depreciates at a rate of 8%, how much will it be worth in 5 years?
To solve this problem, we will use the formula for exponential decay as follows: V = P * e^(-rt) where V is the value after t years, P is the initial value, r is the annual interest rate as a decimal, and t is the time in years.
What is Depreciation: Depreciation is dependent on a number of estimates.The method in which companies determine the depreciation value of their assets is different from one another. Some companies may use a straight line method of depreciation and another may count the depreciation according to asset's production value. What is exponential decay: An exponential function's curve is created by a pattern of data called exponential decay, which exhibits higher decreases over time .Given that a brand new Audi R8 is purchased for $148,700 before taxes, and the car depreciates at a rate of 8%, we can find how much it will be worth in 5 years. Using the formula for exponential decay, we have V = P * e^(-rt) where P = $148,700r = 0.08t = 5. Therefore,V = $148,700 * e^(-0.08 * 5), V = $148,700 * e^(-0.4)V ≈ $82,429.61. Therefore, the car will be worth approximately $82,429.61 in 5 years.
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Stacy's time in her 50-meter freestyle race as measured with a stopwatch was 32. 4 seconds. The more precise electronic touchpad measured her time as 32. 36 seconds. What is the percent error for the stopwatch's measurement?
To find the percent error for the stopwatch's measurement of Stacy's time in her 50-meter freestyle race, we'll use the following formula:
Percent Error = (|(Measured Value - Actual Value)| / Actual Value) * 100
Here, the Measured Value is the stopwatch's time (32.4 seconds), and the Actual Value is the electronic touchpad's time (32.36 seconds).
Step 1: Calculate the absolute difference between the measured and actual values:
|32.4 - 32.36| = 0.04
Step 2: Divide the absolute difference by the actual value:
0.04 / 32.36 = 0.001236
Step 3: Multiply the result by 100 to get the percentage:
0.001236 * 100 = 0.1236%
The percent error for the stopwatch's measurement of Stacy's time in her 50-meter freestyle race is approximately 0.124%.
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The radian measure -1.7 pi is equivalent to -306 degrees.
How does sleep affect memory retention?To find the percent error of the stopwatch's measurement, we need to compare it to the more precise electronic touchpad measurement. The formula for percent error is:
percent error = (|measured value - actual value| / actual value) x 100%
In this case, the measured value is 32.4 seconds, the actual value is 32.36 seconds, and the absolute difference between them is 0.04 seconds. Plugging these values into the formula, we get:
percent error = (|32.4 - 32.36| / 32.36) x 100% = 0.124%
Therefore, the percent error for the stopwatch's measurement is 0.124%. This means that the stopwatch's measurement was very close to the actual value, with an error of only 0.124% of the actual value.
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find the perimeter of the equilateral triangle whose area is 16root3/4
The perimeter of the equilateral triangle whose area is 16root3/4 is 15.9[tex]\sqrt{3/4} cm[/tex]
What is an equilateral triangle?You should understand that a triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted
An equilateral triangle is a special case of an isosceles triangle in which all three sides have the same length
Let the sides of the triangle be a
a + a + a = 16root3/4
3a = 16[tex]\sqrt{3/4}[/tex]
a = 5.3[tex]\sqrt{3/4}[/tex]
Therefore the perimeter of the equilateral triangle is
5.3[tex]\sqrt{3/4} + 5.3\sqrt{3/4} +5.3\sqrt{3/4}[/tex]
Therefore, the perimeter is 15.9[tex]\sqrt{3/4} cm[/tex]
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If the arch were 32 inches wide but 44 inches tall high, how could you modify your function W to model the new arch
This new function W(x) = 22 - (11/8) * (x - 16)² will model the shape of the arch with dimensions of 32 inches wide and 44 inches tall.
To modify the function W to model the new arch with dimensions of 32 inches wide and 44 inches tall, we need to adjust the formula to reflect the new proportions.
Currently, the function W is defined as:
W(x) = h/2 - h/(2a) * (x - a)²
Where h is the height of the arch and a is half of the width of the arch.
To modify the function for the new arch, we need to adjust the value of a to reflect the new width of 32 inches. Since a is half the width, we have:
a = 32/2 = 16
We also need to adjust the value of h to reflect the new height of 44 inches. Therefore, the new function for the arch would be:
W(x) = 44/2 - 44/(2*16) * (x - 16)²
Simplifying this expression, we get:
W(x) = 22 - (11/8) * (x - 16)²
This new function will model the shape of the arch with dimensions of 32 inches wide and 44 inches tall. The parabolic shape of the function will remain the same, but the specific coefficients in the function have been adjusted to reflect the new proportions of the arch.
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Answer the questions below to determine what kind of function is depicted below
Answer:
This function is an exponential function because the base is a constant and the exponent is a variable.
Express the negation of each of these statements in terms of quantifiers without using the negation symbol.
a) ∀x(x > 1)
b) ∀x(x ≤ 2)
c) ∃x(x ≥ 4)
d) ∃x(x < 0)
e) ∀x((x < −1) ∨ (x > 2))
f ) ∃x((x < 4) ∨ (x > 7))
The negation of each of these statements in terms of quantifiers without using the negation symbo
a) There exists at least one x such that x is not greater than 1.
b) There exists at least one x such that x is not less than or equal to 2.
c) For all x, x is less than 4.
d) For all x, x is greater than or equal to 0.
e) There exists at least one x such that either x is not less than or equal to -1 or x is not greater than 2.
f) For all x, x is not less than 4 and x is not greater than 7.
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Abby makes wants to make a gallon of punch. She uses 2 quarts of orange juice 1 cup of lemon juice and 2 1/2 pints of pineapple juice. How many cups of water should you add to make 1 gallon?
Abby wants to make 1 gallon (16 cups) of punch, she will need to add 16 - 14 = 2 cups of water to reach the desired amount.
To answer your question about how many cups of water Abby should add to make 1 gallon of punch, let's first convert all the given measurements to cups. One gallon is equivalent to 16 cups.
1. Orange juice: Abby uses 2 quarts of orange juice. Since there are 4 cups in a quart, she uses 2 x 4 = 8 cups of orange juice.
2. Lemon juice: Abby uses 1 cup of lemon juice.
3. Pineapple juice: Abby uses 2 1/2 pints of pineapple juice. There are 2 cups in a pint, so she uses (2 1/2) x 2 = 5 cups of pineapple juice.
Now, let's add up the cups of orange juice, lemon juice, and pineapple juice: 8 + 1 + 5 = 14 cups. Since Abby wants to make 1 gallon (16 cups) of punch, she will need to add 16 - 14 = 2 cups of water to reach the desired amount.
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Which equations have the same value of x as 3/5 (30 x minus 15) = 72? Select three options.
A. 18 x - 15 = 72
B. 50 x -25 = 72
C. 18 x - 9 = 72
D. 3 (6 x - 3) = 72
E. x = 4.5
The equations that have the same value of x as 3/5 (30 x - 15) = 72 are C, D, and E.
Choosing the equations that are equivalentTo solve for x in 3/5 (30 x - 15) = 72, we can first simplify the left side by distributing the 3/5:
3/5 (30 x - 15) = 18 x - 9
Now we can solve for x by setting the right side equal to 72:
18 x - 9 = 72
Adding 9 to both sides:
18 x = 81
Dividing by 18:
x = 4.5
So we know that option E is one of the correct answers.
To check which of the other options have the same value of x, we can substitute x = 4.5 into each equation and see if it simplifies to 72:
A. 18 x - 15 = 72
18(4.5) - 15 = 72
81 - 15 = 72 (not equivalent)
B. 50 x - 25 = 72
50(4.5) - 25 = 200 - 25 = 175 (not equivalent)
C. 18 x - 9 = 72
18(4.5) - 9 = 72 (equivalent)
D. 3 (6 x - 3) = 72
3(6(4.5) - 3) = 3(24) = 72 (equivalent)
Therefore, the equations that have the same value of x as 3/5 (30 x - 15) = 72 are C, D, and E.
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suppose x is a random variable with mean mu and standard deviation sigma. If a large number of trials are observed, at least what percentage of these values is expected to lie between mu minus 2 sigma and mu plus 2 sigma?
At least 95% of the observed values are expected to lie between mu minus 2 sigma and mu plus 2 sigma.
This is because of the empirical rule, also known as the 68-95-99.7 rule, which states that in a normal distribution, approximately 68% of the observations will fall within one standard deviation of the mean, about 95% of the observations will fall within two standard deviations of the mean, and around 99.7% of the observations will fall within three standard deviations of the mean.
In this case, we are given that x has mean mu and standard deviation sigma. Therefore, about 95% of the values of x are expected to lie between mu minus 2 sigma and mu plus 2 sigma, as this interval covers two standard deviations on either side of the mean.
Mathematically, we can express this as:
P(mu - 2sigma < x < mu + 2sigma) ≈ 0.95
where P is the probability that x falls within the interval mu - 2sigma to mu + 2sigma.
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Help pls! Find the area of the circle
Use π = 3.14 and round your answer to the nearest hundredth.
the area of the circle is 615. 4 m²
How to determine the area
The formula that is used to calculate the area of a circle is expressed with the equation.
We have the equation as;
A = πr²
Such that the parameters are given as;
A is the area of the circleπ takes the constant value of 22/7 or 3.14r is the radius of the circleFrom the diagram shown, we have that;
A = unknown
r = 14m
Now, substitute the values, we get;
Area = 3.14 ×14²
Find the square value
Area = 3.14(196)
Multiply the values
Area = 615. 4 m²
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What is amplitude in Trig.
Answer:
It the distance from mid line to top of wave.
Step-by-step explanation:
Answer:
Height of a wave (from mid line to max.
Step-by-step explanation:
Use a triple integral to find the volume of the solid bounded by the parabolic cylinder y = 2x2 and the planes z = 0,2= 2 and y = 4.
Using the triple integral to find the volume of the solid bounded by the parabolic cylinder is 32/15 cubic units.
The given solid is bounded by the parabolic cylinder y = 2x², the plane z = 0, the plane z = 2, and the plane y = 4.
To find the volume of the solid using a triple integral, we can set up the integral as follows:
∫∫∫E dV
where E is the region of integration in three dimensions.
Region E can be described as:
0 ≤ z ≤ 2
0 ≤ y ≤ 4
0 ≤ x ≤ √(y/2)
Therefore, the triple integral can be written as:
∫0² ∫[tex]0^4[/tex] ∫[tex]0^{\sqrt(y/2)}[/tex] dx dy dz
Evaluating the integral gives us the volume of the solid:
V = ∫0² ∫[tex]0^4[/tex] ∫[tex]0^{\sqrt(y/2)}[/tex] dx dy dz = 32/15
Hence, the volume of the solid is 32/15 cubic units.
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the following table shows the number of miles a hiker walked on a trail each day for 6 days. day 1 2 3 4 5 6 number of miles 8 5 7 2 9 8 what was the mean number of miles the hiker walked for the 6 days? responses 3.5 3.5 4.5 4.5 6.5 6.5 7.5 7.5 8
The mean number of miles the hiker walked for the 6 days was 6.5 miles.
To calculate the mean or average of a set of numbers, we add up all the numbers and then divide the sum by the number of items in the set. In this case, we have the number of miles the hiker walked on each of the six days. To find the total number of miles the hiker walked, we simply add up all the numbers
8 + 5 + 7 + 2 + 9 + 8 = 39
Next, we divide the total number of miles by the number of days (which is 6) to get the average or mean number of miles the hiker walked per day:
Mean number of miles = Total number of miles / Number of days
= 39 / 6
= 6.5
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A rectangular prism has a square
base with edge length (x + 1). Its
volume is (x + 1)2(x – 3). What
does the expression (x + 1)(x – 3)
represent?
area of the base
area of one side
height of the prism
surface area of the prism
The expression (x + 1)(x - 3) represents the Area of base of the prism.
What is Prism?a crystal is a polyhedron containing a n-sided polygon base, a respectable halfway point which is a deciphered duplicate of the first, and n different countenances, fundamentally all parallelograms, joining relating sides of the two bases. Translations of the bases exist in every cross-section that runs parallel to the bases.
According to question:
The volume of a rectangular prism is given by the formula V = Bh, where B is the area of the base and h is the height of the prism. In this case, the base is a square with edge length (x + 1), so its area is (x + 1)^2. The volume of the prism is given as (x + 1)^2(x - 3).
We can find the height of the prism by dividing the volume by the area of the base:
B = V/h = (x + 1)^2(x - 3)/(x + 1) = (x + 1)(x - 3)
Therefore, the expression (x + 1)(x - 3) represents the Area of base of the prism.
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Federico enjoys catching pokemons in university campus. One day, while trying to catch charmander, he found the best spot next to a
perfectly circular pond. He was 43 feet from the bank and 75 feet from the point of tangency. Determine the radius of the pond using the
given information. Round to the nearest integer,
The radius of the pond is 32 feet, under the condition that 43 feet from the bank and 75 feet from the point of tangency.
Let us consider that the center of the circle O, the point of tangency T, and Federico's position P.
We can utilize these two points to form a line. The point of tangency is the place where Federico is closest to the pond. The radius of the pond is considered perpendicular to this line and passes through the point of tangency.
Firstly, we have to the distance between Federico's position P and covers passes through points T and B (the bank). This distance is equivalent to the given radius of the circle. We have to apply the formula for the distance between a point and a line to find this distance.
Let us assume this distance as d.
d = (|BT x BP|) / |BT|
Here
|BT| = line segment length of BT,
|BP| = line segment length of BP,
BT x BP = vectors cross product of BT and BP.
Here we evaluate |BT| applying the Pythagorean theorem
|BT|² = 75²+ r²
Here,
r = radius concerning the circle.
Then,
|BP|² = 43² + r²
Staging these values into our formula for d:
d = (|BT x BP|) / |BT|
= (|BT| × |BP|) / |BT|
= |BP|
= √(43² + r²)
We want to solve for r, so we can square both sides:
d² = 43² + r²
r² = d² - 43²
r = √(d² - 43²)
Placing in d = 75,
r = √(75² - 43²)
≈ 32 feet
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In a ABCD Rhombus, B angle minus A equals 20 degrees. What degrees are all the angles of the Rhombus if B-A=20°?
All the angles of the Rhombus if B-A=20 is angle A = angle C = 80°, and angle B = angle D = 100°.
In a rhombus ABCD, if angle B minus angle A equals 20 degrees (B-A=20°), we can find the degree measures of all the angles.
Step 1: Recognize that in a rhombus, opposite angles are equal. Therefore, angle A = angle C and angle B = angle D.
Step 2: Remember that the sum of the angles in any quadrilateral is 360 degrees. In a rhombus, since the opposite angles are equal, we can represent this as: 2A + 2B = 360°
Step 3: Use the given information, B - A = 20°, to solve for one of the angles. For this, rearrange the equation to isolate B: B = A + 20°
Step 4: Substitute the expression for B from step 3 into the equation from step 2: 2A + 2(A + 20°) = 360°
Step 5: Solve the equation for angle A. 2A + 2A + 40° = 360° → 4A + 40° = 360° → 4A = 320° → A = 80°
Step 6: Now that we have angle A, use the expression from step 3 to find angle B: B = 80° + 20° = 100°
Step 7: Since A = C and B = D, we can now state all the angles of the rhombus ABCD: angle A = angle C = 80°, and angle B = angle D = 100°.
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