The predicted total sales for the ice-cream parlor when the average temperature is 72°F is $950.00.
You are asked to find the predicted total sales (s) for the ice-cream parlor when the average temperature (t) is 72°F, using the trend line equation s = 12.75t + 32.
Step 1: Plug the given temperature (72°F) into the trend line equation:
s = 12.75(72) + 32
Step 2: Calculate the value of 12.75(72):
12.75 * 72 = 918
Step 3: Add 32 to the result from Step 2:
918 + 32 = 950
So, the predicted total sales for the ice-cream parlor when the average temperature is 72°F is $950.00. Therefore, the correct answer is (a) $950.00.
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Determine if the expression zx^3/9-x^3 is a polynomial or not. if it is a polynomial, state the type and degree of the polynomial.
This expression is not a polynomial, and it doesn't have a type or degree.
The expression zx^3/9-x^3 can be simplified as:
zx^3/(9-x^3)
This expression is not a polynomial because it contains a variable (x) in the denominator, which makes it a rational expression.
A polynomial is an expression of one or more terms involving only constants and variables raised to positive integer powers, with no variables in the denominators.
Therefore, this expression is not a polynomial, and it doesn't have a type or degree.
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if there are 40% math's books in school library containing 1800 books in total find the number of the math's books
Answer: 720
Step-by-step explanation: 1800 x 40%
Answer:
720 math books
Step-by-step explanation:
1800×40%= 720
there are 720 math books
What is the value of x log3 x=4
Answer:
x=81
Step-by-step explanation:
Rewrite log_3 (x)=4 in exponential form using the definition of a logarithm. If x and b are positive real numbers and b≠1, then log_b(x)=y is equivalent to b^y=x.
Rewrite the equation as x=3^4
Raise 3 to the power of 4
x=81
For Items 6-10, the height of an object, in centimeters, is modeled by the function y = 42sin (π/10 (x-h)+ 55. Determine whether each statement is always, sometimes, or never true.
6. The period of the function is 20.
7. The maximum height of the object is 55 centimeters
8. The minimum height of the object occurs when x=0
9. The graph of the function has the midline y= 55
10. The amplitude of the function is 84.
Answer:
It's fascinating to observe how the volume of different shapes can vary based on their measurements. For instance, a cylinder with a height of 6 centimeters and radius r1 has a volume of 302 cubic centimeters. Do you require further assistance?
As for the new set of instructions, please consider the following statements:
6. Sometimes true. The period of the function is determined by the formula T= 2π/b, where b is the coefficient of x in the argument of the sine function. In this case, b = π/5, so T= 10.
7. Always true. The maximum height of the object is equal to the amplitude of the function plus the vertical shift, which is 55 centimeters.
8. Sometimes true. The minimum height of the function occurs when the sine function has a value of -1, which happens at x= h-5. So, if h= 0, then x= -5, which means the statement is sometimes true depending on the value of h.
9. Always true. The midline of the function is determined by the vertical shift, which is 55 in this case.
10. Always true. The amplitude of the function is given by A= |b|, where b is the coefficient of x in the argument of the sine function. In this case, A= 42π/5, which simplifies to 84.
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Area of parallelograms, quadrilaterals and polygons - tutorial - part 2. level f
ft
what is the area of the first triangle?
1 ft
1ft
1ft
3ft
4ft
The area of the first triangle is 0 square feet.
To find the area of the first triangle with the given side lengths of 1 ft, 3 ft, and 4 ft, you can use Heron's formula.
Calculate the semi-perimeter (s) of the triangle:
s = (a + b + c) / 2
where a, b, and c are the side lengths of the triangle.
s = (1 + 3 + 4) / 2 = 8 / 2 = 4 ft
Apply Heron's formula to find the area (A) of the triangle:
A = √(s * (s - a) * (s - b) * (s - c))
A = √(4 * (4 - 1) * (4 - 3) * (4 - 4))
A = √(4 * 3 * 1 * 0)
A = √0 = 0
The area of the first triangle is 0 square feet. This means that the given side lengths do not form a valid triangle, as two sides' lengths (1 ft and 3 ft) do not add up to be greater than the length of the third side (4 ft).
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PLEASE HELP ASAP WILL GIVE BRAINLIEST
Lizzie came up with a divisibility test for a certain number m≠1: Break a positive integer n into two-digit chunks, starting from the ones place. (For example, the number 354764 would break into the two-digit chunks 35, 47, and 64) Find the alternating sum of these two-digit numbers, by adding the first number, subtracting the second, adding the third, and so on. (In our example, this alternating sum would be ) Find m and show that this is indeed a divisibility test for m (by showing that n is divisible by m if and only if the result of this process is divisible by m )
The value of m is 11, and the divisibility test states that n is divisible by 11 if and only if the alternating sum of its two-digit chunks is divisible by 11.
How to prove the divisibility test?Let's consider the given divisibility test proposed by Lizzie. The process involves breaking a positive integer, n, into two-digit chunks and finding the alternating sum of these chunks. The alternating sum is obtained by adding the first number, subtracting the second, adding the third, and so on.
To find the value of m that makes this a divisibility test, we need to analyze the properties of this test. Let's assume that n is divisible by m.
When n is divisible by m, each two-digit chunk in n will also be divisible by m. This means that the alternating sum of these chunks will also be by m since adding or subtracting multiples of m will not change its divisibility.
Conversely, if the alternating sum of the two-digit chunks is divisible by m, it implies that each chunk is divisible by m. Therefore, if the chunks are divisible by m, the original number n will also be divisible by m.
Hence, this process indeed serves as a divisibility test for m, where n is divisible by m if and only if the result of the alternating sum of the two-digit chunks is divisible by m.
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One tire manufacturer claims that his tires last an average of 44000 miles with a standard deviation of 7650 miles. A random sample of 120 of his tires is taken. What is the probability that the average of this sample of tires will last longer than 45000 miles
The probability of a randomly selected tire from this sample having a lifespan greater than 45,000 miles is approximately 0.0639 or 6.39%.
We can use the central limit theorem to approximate the distribution of the sample mean.
In this case, the population mean is 44,000 miles and the population standard deviation is 7,650 miles. We are taking a sample of 120 tires, so the standard deviation of the sample mean is:
σ/√n = 7,650/√120 = 698.68
To find the probability that the sample mean will be longer than 45,000 miles, we need to standardize the sample mean using the formula:
z = (x - μ) / (σ / √n)
where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Plugging in the values, we get:
z = (45,000 - 44,000) / (7,650 / √120) = 1.527
We can then look up the probability corresponding to a z-score of 1.527 in a standard normal distribution table or using a calculator. The probability of a randomly selected tire from this sample having a lifespan greater than 45,000 miles is approximately 0.0639 or 6.39%.
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find the exact area of a square with a diagonal of 8 inches
Answer:
(8/√2)^2 = 64/2 = 32 square inches
Marcia makes jewelry to sell at the artists' fair. She spends $120 to rent a stall at
the fair for the day and each piece of jewelry costs Marcia $15 in materials.
Which equation would compute the number of pieces of jewelry Marcia must sell at
the artists' fair such that the average cost per piece of jewelry would be $20?
110
The equation that can be used to compute the number of pieces of jewellery is ($120 + $15q)/q = $20 .
What is the equation?
Average cost is total cost divided by the quantity of jewellery sold.
Average cost = total cost / quantity
Total cost is the sum of fixed cost and variable cost.
Total cost = fixed cost + variable cost
The fixed cost is the cost of renting the stall. The variable cost is the cost of the materials.
Total cost = $120 + ($15 x q)
T = $120 + $15q
Average cost = ($120 + $15q)/q = $20
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The city is planning a concert that is expected to bring in a crowd of about
200,000 people. The concert will be held in a public park. The city planners are thinking
about the size and shape of the space that will be needed to accommodate this
number of people.
At a much smaller yet similar event, the crowd was estimated to be about
22,000 people. At this event, the crowd was confined to an area that was roughly the
shape of a right triangle with side lengths that were approximately 300 feet, 350 feet,
and 461 feet.
Determine the appropriate dimensions of a similar space with 200,000 people.
Show your work or explain your modeling.
hallar larger
The dimensions of the larger space would be roughly 300 x 350 x 461 feet multiplied by the scaling factor of 3.01. This gives dimensions of approximately 903 x 1053 x 1388 feet.
To determine the appropriate dimensions of a space that can accommodate 200,000 people, we can use the concept of similarity.
We know that the smaller event had a crowd of 22,000 people and the area was roughly a right triangle with side lengths of 300, 350, and 461 feet. We can use the ratio of the number of people to the area to find the scaling factor.
The area of the triangle is (1/2) x 300 x 350 = 52,500 square feet.
The ratio of people to area is 22,000/52,500 = 0.42 people per square foot.
To accommodate 200,000 people, we need an area of 200,000/0.42 = 476,190.5 square feet.
Assuming we maintain the same shape and proportions, we can use the area of the triangle as a guide to find the dimensions of the larger space. Let x be the scaling factor. Then:
(1/2) x (300x) x (350x) = 476,190.5
52,500x² = 476,190.5
x² = 9.05
x = 3.01
In summary, we can use the ratio of people to area to determine the appropriate dimensions of a space that can accommodate 200,000 people. By maintaining the same shape and proportions of a smaller event, we can find the scaling factor needed to determine the dimensions of the larger space.
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Find the surface area of the triangular prism. The base of the prism is an isosceles triangle.
The surface area of the triangular prism is 3152 square cm if the base of the prism is an isosceles triangle.
What exactly is a triangular prism?
When a prism has three rectangular sides and two triangular bases, the prism is said to be triangular. It's a pentahedron. A right triangular prism has two faces and three rectangular sides. Bases refers to the triangle faces, whereas laterals refers to the rectangular sides.
We have a triangular prism, is showing in the picture:
Here a = 25
b = 25
c = 14
h = 44
The surface area of the triangular prism is given by
A = ah + bh + ch + 1/2√-a⁴ + 2(ab)² + 2(ac)² + -b⁴ + 2(bc)² - c⁴
Plug all the values in the formula we get:
A = 3152 square cm
Thus, the surface area of the triangular prism is 3152 square cm if the base of the prism is an isosceles triangle.
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Suppose the area of a trapezoid is 126 yd?. if the bases of the trapezoid are 17 yd and 11 yd long, what is the height?
a 4.5 yd
b. 9 yd
c. 2.25 yd
d. 18 yd
The height of the trapezoid is 9 yards. Therefore, the correct answer is option b. 9 yd.
To find the height of the trapezoid with the given area and base lengths, we will use the formula for the area of a trapezoid:
Area = (1/2) * (base1 + base2) * height
Here, the area is given as 126 square yards, base1 is 17 yards, and base2 is 11 yards. We need to find the height.
1. Substitute the given values into the formula:
126 = (1/2) * (17 + 11) * height
2. Simplify the equation:
126 = (1/2) * 28 * height
3. To isolate the height, divide both sides by (1/2) * 28:
height = 126 / ((1/2) * 28)
4. Calculate the result:
height = 126 / 14
height = 9
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Evaluate the following limit. Use l'Hôpital's Rule when it is convenient and applicable. lim sin 3 X00 lim 3 Xod in ) - 0 () (Type an exact answer.) X
The overall limit is undefined as the the second limit is undefined
The given limit is of the indeterminate form 0/0 and hence we can apply l'Hôpital's Rule to evaluate it.
Applying l'Hôpital's Rule, we get:
lim sin(3x) / (3x) = lim [cos(3x) * 3] / 3 = cos(3x)
Now, we need to evaluate lim (3x)/(1 - cos(x)) as x approaches 0.
Again, this limit is of the indeterminate form 0/0, so we can apply l'Hôpital's Rule once again:
lim (3x)/(1 - cos(x)) = lim (3)/(sin(x)) = 3/0 (which is undefined)
Since the second limit is undefined, the overall limit is also undefined.
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5)
The population of Sanibel, Florida can
be modeled by P = 6191 · 1. 05t,
where t is the number of years since
2016. What was the population in
2016? What percent did the
population increase by each year?
The population increased by a percentage of 5%.
The population of Sanibel, Florida in 2016 can be determined using the given population model P = 6191 * 1.05^t, where t represents the number of years since 2016. To find the population in 2016, we set t to 0 since there are 0 years since 2016.
Step 1: Set t to 0 in the equation:
P = 6191 * 1.05^0
Step 2: Calculate the population P:
P = 6191 * 1
P = 6191
So, the population in Sanibel, Florida in 2016 was 6,191.
Regarding the percent population increase each year, the given model uses an exponential growth formula with a constant factor of 1.05. The factor (1.05) represents a 5% increase in the population each year.
In summary, the population in Sanibel, Florida in 2016 was 6,191, and the population increases by 5% each year. This exponential growth model demonstrates how the population continues to grow at a steady rate, contributing to the overall population increase in the area.
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The function f(t)=350(1. 2)^{365t}f(t)=350(1. 2)
365t
represents the change in a quantity over t years. What does the constant 1. 2 reveal about the rate of change of the quantity?
The constant 1.2 in the function represents the rate of change of the quantity per year.
Specifically, it represents the factor by which the quantity grows or decays over a year.
Since 1.2 is greater than 1, the function describes an exponential growth, where the quantity is multiplied by 1.2 each year.
For example, after one year, the quantity is multiplied by 1.2, after two years it is multiplied by 1.2 squared, and so on.
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please help me answer this (can give brainliest)
a) The graph of the given lines is as attached
b) The area of the enclosed triangle is: 8 square units
How to graph linear equations?The general form of expression of linear equations in slope intercept form is expressed as:
y = mx + c
where:
m is slope
c is y-intercept
We are given the equations as:
y = x + 5
y = 5
x = 4
The graph of these three linear equations is as shown in the attached file
2) The area of the given triangle enclosed by the three lines is gotten from the formula:
A = ¹/₂ * b * h
where:
A is area
b is base
h is height
Thus:
A = ¹/₂ * 4 * 4
A = 8 square units
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What is the volume, in cubic centimeters, of a cylinder with a height of 5 cm and a base radius of 10 cm, to the nearest tenths place?
The volume of the cylinder is approximately 1570.8 cubic centimeters, rounded to the nearest tenth.
A cylinder is a three-dimensional object with two congruent circular bases that are parallel to each other. The volume of a cylinder can be calculated using the formula V = πr²h, where V is the volume, r is the radius of the base, and h is the height of the cylinder.
In this problem, we are given that the height of the cylinder is 5 centimeters and the radius of the base is 10 centimeters. By substituting these values into the formula, we get:
V = π x 10² x 5
V = 500π
To calculate the volume of the cylinder, we can use an approximation for the value of pi. Taking pi to be approximately 3.14, we can calculate the volume as follows:
V ≈ 500 x 3.14
V ≈ 1570.8
Therefore, the volume of the cylinder to the nearest tenths place is approximately 1570.8 cubic centimeters.
It is important to note that the answer is an approximation since pi is an irrational number with an infinite number of decimal places. However, rounding to the nearest tenths place provides a reasonable level of precision for this calculation.
In summary, the volume of the cylinder is 1570.8 cubic centimeters, and the calculation is based on the given values and the formula for the volume of a cylinder.
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Find the probability that a randomly selected point within the circle falls in the white area. R=4 cm 2. 5 cm 3 cm 3 cm [?]% Round to the nearest tenth of a percent.
The probability is approximately 45.4% (rounded to the nearest tenth of a percent).
To find the probability that a randomly selected point within the circle falls in the white area, we need to find the area of the white region and divide it by the total area of the circle.
The total area of the circle is:
A = πr² = π(4 cm)² = 16π cm²
The area of the white region can be found by subtracting the area of the two semicircles from the area of the circle:
White area = A - 2(1/2π(2.5 cm)²) = 16π - 2(4.375π) = 7.25π cm²
So, the probability that a randomly selected point within the circle falls in the white area is:
P(white area) = (white area)/(total area) = (7.25π)/(16π) ≈ 45.4%
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What is the vertex of the graph of the equation Y=3x2( to the second power) +6x+1
A.(-1,-2)
B.(-1, 10)
C.(1, -2)
D.(1, 10)
Answer: To find the vertex of the graph of the equation Y=3x^2+6x+1, we can use the formula:
x = -b/2a
where a = 3 and b = 6, which are the coefficients of the x^2 and x terms, respectively.
x = -6/(2 x 3) = -1
Substituting x = -1 into the equation, we get:
Y = 3(-1)^2 + 6(-1) + 1 = -2
Therefore, the vertex of the graph is (-1, -2), so the answer is A. (-1,-2).
Step-by-step explanation:
PLS HELP! and actually answer the question please
Step-by-step explanation:
First start with the graph of y = | x|
then shift it RIGHT one unit
| x -1 |
then shift it DOWN one unit
y= |x-1| -1
A candlemaker prices one set of scented candles at $10 and sells an average of 200 sets each week. He finds that when he reduces the price
by $1, he then sells 50 more candle sets each week. A function can be used to model the relationship between the candlemaker's weekly
revenue, R(x) after xone-dollar decreases in price.
R(x)
R(x)
6,000+
4,000+
6,000+
1,000+
2. 000
2,000+
2,000
2,000+
-4,000
4,000
6,000+
-6,000
Graph w
R(x)
Graph X
R(x)
6,000+
1,000+
6,000+
1,000+
2,000+
2,000
-2,000+
2,000
1,000+
4,000
-6,000
6,000
Graph Y
Graph Z
This situation can be modeled by the equation y =
x +
x +
and by graph
Next
The equation of demand of candle sticks can be modeled by
y = 14 - 0.02x while the revenue function will be xy = 14x - 0.02x².
Here we are given the information that
200 candles are sold for $10 and,
250 candles are sold for $9
Let the Price be y while the Quantity sold be x
Hence, by one unit decrease in price P, the quantity sold is increased by 50 units.
Here the slope of the function will be
(10 - 9)/(200 - 250)
= - 1/50
= - 0,02
Now we will use the formula of the equation of a straight line
(y - y₁) = m(x - x₁)
where, m is the slope and x₁ , y₁ are some point on line
Hence we get
(y - 10) = -0.02(x - 200)
or, y - 10 = -0.02x + 4
or, y = 14 - 0.02x
The revenue function will be xy = 14x - 0.02x²
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Correct Question
A candlemaker prices one set of scented candles at $10 and sells an average of 200 sets each week. he finds that when he reduces the price by $1, he then sells 50 more candle sets each week. a function can be used to model the relationship between the candlemaker's weekly revenue, r(x), after one-dollar decrease in price. this situation can be modeled by the equation y =
A child lifts a box up from the floor. The child then carries the box with a constant speed to the other side of the room and puts the box down. How much work does he do on the box while walking across the floor at constant speed? Your answer: 0 J More than 0 J More information is needed to determine the answer
0 J. When the child carries the box at a constant speed, the net force on the box is zero because there is no acceleration. Therefore, the work done by the child on the box is zero. So, the correct option is 0 J.
To determine the amount of work done by the child while walking across the floor at a constant speed, we need to consider the following terms: work, force, and displacement.
Work is the amount of energy transferred when a force is applied over a certain distance. It is calculated as:
Work = Force x Distance x cos(θ)
where Force is the applied force, Distance is the displacement, and θ is the angle between the force and displacement vectors.
When the child carries the box with a constant speed across the room, the force applied is equal to the gravitational force acting on the box (i.e., the weight of the box). However, since the force and displacement are in different directions (the force is acting vertically, while the displacement is horizontal), the angle between the force and displacement is 90 degrees.
Now, we know that cos(90°) = 0. Thus,
Work = Force x Distance x cos(90°) = Force x Distance x 0 = 0 J
So, the work done by the child on the box while walking across the floor at constant speed is 0 J.
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Evaluate the line integral, where C is the given
curve.
C
y3ds,
C: x =
t3, y =
t, 0 ≤ t ≤ 5
Please provide the right answer.
Unfortunately, this integral doesn't have a simple closed-form solution. However, you can use numerical methods or software like Wolfram Alpha or a graphing calculator to approximate the value of the integral.
We have:
y = t and ds = sqrt(9t^4 + 1) dt
So, the line integral becomes:
∫C y^3 ds = ∫0^5 (t^3)(sqrt(9t^4 + 1)) dt
Using the substitution u = 9t^4 + 1, we get du/dt = 36t^3, which means dt = du/36t^3. Also, when t = 0, u = 1 and when t = 5, u = 1126.
Substituting these values and simplifying, we get:
∫C y^3 ds = (1/36) ∫1^1126 (u-1/4)(1/2) du
= (1/72) [(u-1)^2 u^(1/2)]_1^1126
= (1/72) [(1125)^2 (1126^(1/2)) - (1)^2 (1^(1/2))]
= 3555.89 (approx)
Therefore, the line integral is approximately equal to 3555.89.
To evaluate the line integral along the curve C with the given parameterization x = t^3 and y = t for 0 ≤ t ≤ 5, we need to find the integral of y^3ds. First, we need to find the derivative of the parameterization with respect to t:
dx/dt = 3t^2
dy/dt = 1
Now, we can find the differential arc length ds, which is given by the formula:
ds = √((dx/dt)^2 + (dy/dt)^2) dt
ds = √((3t^2)^2 + (1)^2) dt
ds = √(9t^4 + 1) dt
Next, substitute the parameterization of y in terms of t (y = t) into the integral:
∫(y^3 ds) = ∫(t^3 √(9t^4 + 1)) dt, with limits 0 to 5.
Now, evaluate the integral:
∫(t^3 √(9t^4 + 1)) dt from 0 to 5.
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What type of angles are 3 and 6
A. Alternate interior angles
B. Alternate exterior angles
C. Supplementary angles
D. Vertical angles
If the minimum perimeter of a quadrilateral is 200cm, what is the maximum area of the quadrilateral
guesses will be reported
first correct answer gets brainliest <3
The maximum area of the quadrilateral is achieved when it is a square with a side length of 50cm, resulting in an area of 2500cm².
This is because a square has equal sides and equal angles,
resulting in the most efficient use of the perimeter to enclose the maximum area.
To see why, consider a rectangle with a perimeter of 200cm.
If the rectangle is long and thin, with one side much longer than the others,
then it will have a smaller area than a square with the same perimeter.
This is because the longer sides of the rectangle will be less effective at enclosing area than the shorter sides.
Hence, The maximum area of a quadrilateral with a minimum perimeter of 200cm is achieved when the quadrilateral is a square.
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Let z be a random variable with a standard normal distribution. Find the indicated probability. (Enter a number. Round your answer to four decimal places. )
P(z ≥ 1. 41) =
If you let z be a random variable with a standard normal distribution, the indicated probability is 0.0793. So, the probability P(z ≥ 1.41) = 0.0793
To find the probability P(z ≥ 1.41) for a standard normal distribution, you can use a Z-table or calculator to find the area to the right of z = 1.41.
Using a Z-table or calculator, you will find the value of P(z ≥ 1.41) is approximately 0.0793. So, the probability P(z ≥ 1.41) = 0.0793. Remember to round your answer to four decimal places.
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If x=2, y=4,m=-1 and n=3, find the value of x^m+n * y^n-m/x^m-n * y^n+m
Answer:
1024
Step-by-step explanation:
I hope everything I wrote is clear, I really need to sharpen my pencil oof
The radius of a circle is 7 centimeters. What is the circumference. Round the answer to the nearest hundredth
Answer:
43.96 cm
Step-by-step explanation:
Given
Radius ( r ) = 7 cm
To find : Circumference of Circle
Formula
Circumference of Circle = 2πr
Note
The Value of π = 3.14
Circumference of Circle
= 2πr
= 2 × 3.14 × 7
= 43.96 cm
Answer:
Not Rounded: 43.9822971503
Rounded: 44
Step-by-step explanation:
r= radius
2[tex]\pi[/tex]r
2[tex]\pi[/tex](7)
= 43.9822971503
Question 11 Σ Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If it diverges to infinity, state your answer as "oo" (without the quotation marks). If it diverges to negative infinity, state your answer as "-00". If it diverges without being infinity or negative infinity, state your answer as "DNE". 10 10 dc 2 6
The given integral is ∫₁₀ (10/(x-2)) dx.
The given integral is divergent and the answer is "oo".
To see why the integral is divergent, we can use the following limit comparison test:
∫₁₀ (10/(x-2)) dx ~ ∫₁₀ (1/x) dx, as x → 2.
The symbol "~" means "is asymptotic to". We can use this comparison because as x approaches 2 from the right, the function 10/(x-2) approaches positive infinity.
Now, we know that the integral ∫₁₀ (1/x) dx diverges because the improper integral of 1/x from 1 to infinity diverges. Therefore, by the limit comparison test, the original integral is also divergent.
Hence, the given integral ∫₁₀ (10/(x-2)) dx is divergent, and the answer is "oo".
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The Environmental Protection Agency has determined that safe drinking water should have an average pH of 7. Water is unsafe if it deviates too far from 7 in either direction.You are testing water from a new source and randomly select 30 vials of water. The mean pH level in your sample is 6.4, which is slightly acidic.The Standard deviation of the sample is 0.5.(a) Does the data provide enough evidence at a = 0.05 level that the true mean pH of water from this source differs from 7?(b) A 95% confidence interval for the true mean pH level of the water is (6.21, 6.59). Interpret this interval.(c) Explain why the interval in part (b) is consistent with the result of the test in part (a).
a. The data provided enough evidence at a = 0.05 level that the true mean pH of water from this source differs from 7
b. A 95% confidence interval for the true mean pH level of the water is (6.21, 6.59) means about 95% of those intervals would contain the true mean pH level.
c. The estimated mean pH level of seven is not included in the interval in section (b). This is consistent with the result of the test in part (a), which also rejects the null hypothesis that the true mean pH level is 7.
(a) To test whether the true mean pH of water from this source differs from 7, we can perform a one-sample t-test. The null hypothesis is that the true mean pH is equal to 7, and the alternative hypothesis is that the true mean pH is not equal to 7.
The test statistic can be calculated as follows:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
t = (6.4 - 7) / (0.5 / sqrt(30))
t = -3.07
Using a t-table with 29 degrees of freedom at a significance level of 0.05 (two-tailed test), the critical t-value is ±2.045. Since the calculated t-value (-3.07) is outside of the critical t-value range, we can reject the null hypothesis and conclude that there is enough evidence at a = 0.05 level to suggest that the true mean pH of water from this source differs from 7.
(b) A 95% confidence interval for the true mean pH level of the water is (6.21, 6.59). This means that if we were to take many random samples of size 30 from this water source, and construct a 95% confidence interval for each sample mean pH level, then about 95% of those intervals would contain the true mean pH level.
(c) The interval in part (b) does not include the hypothesized mean pH level of 7. This is consistent with the result of the test in part (a), which also rejects the null hypothesis that the true mean pH level is 7.
The confidence interval provides additional information by giving a range of plausible values for the true mean pH level, and we can see that all of the values in this range are below 7, indicating that the water is indeed slightly acidic.
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