Gus is designing a cylinder to ship liquids using the constraints given.
The inside of the cylinder must hold from 475 to 480 cubic centimeters of
liquid.
The diameter must be at least 8 centimeters and at most 10 centimeters.
What are a possible radius and corresponding height, in centimeters, for the inside
of a cylinder that meets the constraints? Round the answers to the nearest tenth.
You will focus your answer on the lower constraints of 475 volume and diameter of 8
cm for our take on this problem. You will fill in the height you find rounded to the
nearest tenth with no cm. Use 3.14 for pi.
Answer:
The possible radius and height for the inside of the cylinder that meets the constraints are r = 4 cm and h = 9.4 cm.
Step-by-step explanation:
The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height. We want to find a radius and height that will give us a volume between 475 and 480 cubic centimeters, with a diameter between 8 and 10 centimeters. First, we’ll use the lower limits of the constraints: a volume of 475 cubic centimeters and a diameter of 8 centimeters. The diameter is 8 centimeters, so the radius is 4 centimeters. We can plug in these values to the formula for volume and solve for h: 475 = 3.14 x 4^2 x h 475 = 50.24h h = 9.44 So a possible radius and height for the inside of the cylinder that meets the constraints are: r = 4 cm and h = 9.4 cm.
a cathedral ceiling shown in the figure below is 8 feet high at the west wall of a room. as you go from the west wall toward the east wall, the ceiling slants upward. three feet from the west wall, the ceiling is 10.5 feet high. exercise (a) what is the slope of the ceiling? step 1 the slope is the rise over the run. when we move 3 feet east from the west wall the run is incorrect: your answer is incorrect. feet. the corresponding rise is the difference in ceiling heights. rise
The slope of the cathedral ceiling is 0.83.
To calculate the slope of the cathedral ceiling, we need to find the rise over the run. The rise is the difference in ceiling heights between the two points. The run is the horizontal distance between the two points.
From the problem statement, we know that the ceiling is 8 feet high at the west wall and 10.5 feet high three feet from the west wall. So the rise is:
rise = 10.5 - 8 = 2.5 feet
The run is the horizontal distance between the two points, which is:
run = 3 feet
Now we can calculate the slope:
slope = rise/run = 2.5/3 = 0.83
Therefore, the slope of the cathedral ceiling is 0.83.
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Ethan had some solid-coloured socks and patterned socks. He had \frac{3}{4} as many solid-coloured socks as patterned socks. He threw away 16 Paris of solid-coloured socks and 16 Paris of patterned socks. \frac{1}{5} of his socks were now solid-coloured socks. How many pairs of patterned socks did Ethan have at first?
If ethan threw away 16 Paris of solid-coloured socks and 16 Paris of patterned socks, Ethan had 140 pairs of patterned socks at first.
Let's start by assuming that Ethan had x pairs of patterned socks. According to the problem statement, Ethan had 3/4 as many solid-coloured socks as patterned socks. Therefore, the number of solid-coloured socks he had can be represented as 3/4*x.
Ethan threw away 16 pairs of solid-coloured socks and 16 pairs of patterned socks. So, after the clean-up, he had (3/4*x)-16 pairs of solid-coloured socks and (x-16) pairs of patterned socks.
The problem states that 1/5 of his socks were now solid-coloured socks. So, we can set up an equation:
(3/4x-16) = (1/5)(3/4*x + x - 32)
Simplifying and solving for x, we get:
x = 140
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The radius of a circle is 9 millimeters. What is the circle's circumference?
Use 3.14 for л.
Answer:
56.55
Step-by-step explanation:
C=2πr=2·π·9≈56.54867
Suppose that water is poured into a tank at a rate of 2000t 1000 gallons per minute for t > 0. If the tank started with 5000 gallons of water how much water is in the tank after 4 minutes
There is a volume of 14,000 gallons of water in the tank after 4 minutes.
To answer your question, let's consider the given terms: the rate at which water is poured into the tank (2000t + 1000 gallons per minute) and the initial volume of water in the tank (5000 gallons).
Since the rate is given in terms of time t, we can find the volume of water poured in after 4 minutes by plugging t = 4 into the rate equation:
Volume poured in 4 minutes = 2000(4) + 1000
Volume poured = 8000 + 1000
Volume poured = 9000 gallons
Now, add this to the initial volume of water in the tank:
Total water in tank after 4 minutes = Initial volume + Volume poured
Total water = 5000 + 9000
Total water = 14,000 gallons
So, after 4 minutes, there are 14,000 gallons of water in the tank.
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There is a spinner with 14 equal areas, numbered 1 through 14. If the spinner is spun one time, what is the probability that the result is a multiple of 3 or a multiple of 4?
Answer:
Okay, let's solve this step-by-step:
There is a spinner with 14 equal areas, numbered 1 through 14
We want to find the probability that the result is a multiple of 3 or a multiple of 4
There are 14 possible outcomes (numbers 1 through 14) when the spinner is spun.
Of these 14 numbers:
4 are multiples of 3: 3, 6, 9, 12
4 are multiples of 4: 4, 8, 12, 16
However, 12 is also a multiple of both 3 and 4, so we have counted it twice.
We need to subtract 1 from each to account for this:
Multiples of 3: 3
Multiples of 4: 4
So there are 3 possible multiples of 3 and 3 possible multiples of 4.
In total, there are 3 + 3 = 6 possible multiples of 3 or 4.
To calculate probability:
Probability = (Number of favorable outcomes) / (Total possible outcomes)
= 6 / 14
= 3/7
Converting to a percent: 3/7 = 42.9%
Rounded to the nearest whole percent: 43%
Therefore, the probability that the result is a multiple of 3 or a multiple of 4 is 43%.
Step-by-step explanation:
Find the value of x. If a segment looks like a tangent, it is a tangent.
The value of x in the circle is 19.5 .
What is secant line of circle?
A secant is a straight line that intersects a circle at least twice in different places. We can draw an endless number of secants on a circle since a circle has an infinite number of points around its perimeter.
Here the given circle using formula for secant line ,
(A+B).B = (C+D).D
Here A = x , B = 8 , C= 12 and D=10. Then,
=> (x+8).8= (12+10).10
=> 8x+64 = 22*10
=> 8x+64 = 220
=> 8x = 220-64 = 156
=> x = 156/8 = 19.5
Hence the value of x in circle is 19.5.
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solve for x
khan academy solve similar triangles advanced
Value of side CD = x=3
What is similarity of two triangle ?When two triangles are are referred to as similar figures when they share the same shape but different in size.
given that CB and ED are perpendicular to AD,
in triangle ABC and ADE
∠A=∠A (common angle )
∠B=∠D (both are 90 degree)
ΔABC≈ ΔADE by angle-angle similarity .
and we know that when two triangle are similar then their corresponding sides are with in the same ratio or proportional.
so ,here we proved that triangle ABC and ADE are similar,
then ,
[tex]\frac{BC}{ED}=\frac{AB}{AD}[/tex]
[tex]\frac{x}{5}=\frac{9}{9+6}[/tex]
[tex]\frac{x}{5} =\frac{9}{15}\\ X=3[/tex]
x=3
from above result ,value of side CB =x=3
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show that,for all value for x
(2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2
Answer:
Therefore, the answer is: (2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2
Step-by-step explanation:
To prove that (2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2 for all values of x, we can simply expand the left-hand side of the equation and simplify it to match the right-hand side.
Expanding the left-hand side using the distributive property, we get:
(2x-1)(x+2)(3x-1) = (2x^2+3x-2)(3x-1)
= 6x^3 + 7x^2 - 9x + 2
This matches the right-hand side of the equation, so we have proven that (2x-1)(x+2)(3x-1)=6x^3+7x^2-9x+2 for all values of x.
Find the area of the sector in a circle whose radius is 6 and the angle measure is 140 degrees. Round your answer to the nearest hundredth.
Thus, the obtained area of sector for the given circle is found as 43.96 in².
Define about the circle's sector:Two radii that meet at the centre to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle assessment and radius measurement are both crucial for solving circle-related difficulties.
The curved portion that runs along the circle's perimeter and joins the ends of a two radii that make up a sector is known as the sector arc.
given data:
Central angle Ф = 140°radius of circle r = 6 inFormula for the area of sector:
area of sector = Ф /360 * (πr²)
area of sector = 140/360 * (3.14 *6²)
area of sector = 7/18 * 3.14 *36
area of sector = 43.96 in²
Thus, the obtained area of sector for the given circle is found as 43.96 in².
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Before working through each problem, identify the principal, rate, time. Work with your shoulder-partner to find the solution. • Mr. Jackson deposited $1,250 in a new account at his bank. • The bank pays 3.5% simple interest • Mr. Jackson pays no additional deposits or withdrawals. o What amount is closest to the balance of the account at the end of 2 years?
Thus, the amount of the money in the bank account at the end of 2 years is found to be: $1337.5.
Explain about the term simple interest:Simple interest denotes interest that is simply charged on the principal amount, which is the original amount borrowed or deposited. The interest charge is going to be applied once, regardless of how frequently it is applied. Many loans base their calculations on simple interest, but you should double-check before signing anything.
Given data:
Principal P = $1,250Simple Interest rate R = 3.5%Time T = 2 yearsFormula for the estimating simple interest:
SI = PRT/100
SI = 1250*3.5*2 / 100
SI = 87.5
Amount = principal + simple interest
A = 1250 + 87.5
A = 1337.5
Thus, the amount of the money in the bank account at the end of 2 years is found to be: $1337.5.
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(1 point) a spherical balloon is inflated so that its volume is increasing at the rate of 2.8 ft3/min . how rapidly is the diameter of the balloon increasing when the diameter is 1.5 feet?
The diameter of the balloon is increasing at a rate of approximately 0.79 ft/min when the diameter is 1.5 feet.
We can use the formula for the volume of a sphere to relate the rate of change of volume with the rate of change of diameter
V = (4/3)πr^3 = (1/6)πd^3,
where V is the volume, r is the radius, and d is the diameter.
Taking the derivative with respect to time t, we get
dV/dt = (1/2)πd^2 (dd/dt),
where dd/dt is the rate of change of diameter.
We are given that dV/dt = 2.8 ft^3/min and d = 1.5 ft, so we can solve for dd/dt
dd/dt = (2dV/dt)/(πd^2) = (2(2.8))/(π(1.5)^2) ≈ 0.79 ft/min.
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A credit card starts new customers at a $2,000 limit when they are approved for a card. The company adds $500 annually to this limit for customers who pay their bill on time. Choose the equation below that gives the credit limit, Ln, of customers who have payed on time every year, and who are in their nth year of having the card. Then, use this equation to find the credit limit of a customer in their 10th year of having the card.
On solving the provided question we can say that As a result, the credit equation limit of a client who has paid on time every year for the past ten years is $7,000.
What is equation?A mathematical equation is a formula that links two statements and uses the equals sign (=) to indicate equality. In algebra, an equation is a statement that demonstrates the equality of two mathematical expressions. The equal sign divides the variables 3x + 5 and 14 in the equation 3x + 5 = 14, for instance.
The relationship between the two sentences that are located on opposite sides of a letter is explained by a mathematical formula. Frequently, the symbol and the single variable are identical. like in 2x - 4 = 2, for example.
The following equation determines the credit limit, Ln, of customers who have made on-time payments each year and are in the nth year of card ownership:
Ln = $2,000 + $500n
where Ln stands for the credit limit in the nth year and n is the number of years the cardholder has had it.
In the equation above, we substitute n=10 to get a customer's credit limit after ten years of card use:
L10 = $2,000 + $500(10)
L10 = $2,000 + $5,000
L10 = $7,000
As a result, a client with a ten-year history of on-time payments has a credit limit of $7,000.
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Choose the correct phrase in the piece.
Jesus necesita una tapa para un frasco.El perimetro del frasco es 21.98 cm ¿cuanto mide el radio de la tapa?
por favor ocupo ayuda
The radius of the lid would be 3. 5 cm.
How to find the radius ?To estimate the radius of the lid, we must first determine the circumference of the jar, which is equal to the jar's perimeter. The specified circumference is 21.98 cm.
The circumference is:
C = 2πr
We can solve for circumference as:
21. 98 = 2 πr
r = 21. 98 / ( 2 π )
r = 21. 98 / (2 x 3. 1416 )
r = 21. 98 / 6. 2832
r = 3. 5 cm
In conclusion, the radius of the lid is approximately 3.5 cm.
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Anthony has 35 m of fencing to build a three-sided fence around a rectangular plot of land that sits on a riverbank. (The fourth side of the enclosure would be the river.) The area of the land is 132 square meters. List each set of possible dimensions (length and width) of the field.
The two possible sets of dimensions for the rectangular plot are:
Length = 27 meters, Width = 4 meters
Length = 4 meters, Width = 15.5 meters
Let's denote the length of the rectangular plot by L and the width by W. We know that the total length of the fencing needed is 35 m, which we can use to create an equation:
L + 2W = 35
The area of the land is 132 square meters, which we can use to create another equation:
LW = 132
We can solve the first equation for L:
L = 35 - 2W
Substituting this into the second equation, we get:
(35 - 2W)W = 132
Expanding and rearranging, we get a quadratic equation:
2W^2 - 35W + 132 = 0
We can solve for W using the quadratic formula:
W = [35 ± √(35^2 - 4(2)(132))] / (2(2))
W = [35 ± √(841)] / 4
W = [35 ± 29] / 4
Solving for W, we get two possible values:
W = 4 or W = 15.5
If W is 4 meters, then L is:
L = 35 - 2W = 27
If W is 15.5 meters, then L is:
L = 35 - 2W = 4
Therefore, the two possible sets of dimensions for the rectangular plot are:
Length = 27 meters, Width = 4 meters
Length = 4 meters, Width = 15.5 meters
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Question 10(Multiple Choice Worth 2 points)
(11.01 LC)
What is the range of this data set?
The range of this data set include the following: D. 6.
What is a range?In Mathematics and Geometry, a range is the set of all real numbers that connects with the elements of a domain.
Furthermore, the vertical extent of any graph of a function represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.
By critically observing the graph (dot plot) shown in the image attached above, we can reasonably and logically deduce the following range:
Range = {18, 19, 20, 21, 22, 24} = 6.
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What is the name corresponding to the metric symbol mL?
a lot is in the shape of a right triangle. the shorter leg measures 150 m. the hypotenuse is 50 m longer than the length of the longer leg. how long is the longer leg?
Answer:
Starting with the 3-4-5 right triangle, multiply all lengths by 50, obtaining 150, 200, and 250. So the length of the longer leg is 200 meters.
find circle the middle of this equation
Answer: (-15, 20)
Step-by-step explanation:
we can see that the equation is the standard form of a circle:
(X-h)
^2 + (y-k)r^2
Where (h,k) is the center of the circle and r is the radius
H= -15
K= 20
R^2= 100
On average, a certain kind of kitchen appliance requires repairs once every four years. Assume that the times between repairs are exponentially distributed. What is the probability that the appliance will work at least six years without requiring repairs?
The probability that the appliance will work at least six years without requiring repairs is 0.2231 or 22.31%.
To find the probability that the appliance will work at least six years without requiring repairs, we need to consider the exponential distribution and the given average repair time. Given that the appliance requires repairs once every four years on average, the rate parameter (λ) for the exponential distribution is 1/4, or 0.25.
Here we want to find the probability that the appliance will work at least six years without requiring repairs, which can be represented as [tex]P(X ≥ 6)[/tex], where X is the time between repairs. Using the complementary probability, we can rewrite this as [tex]P(X ≥ 6) = 1 - P(X < 6)[/tex]
The cumulative distribution function (CDF) of the exponential distribution is given by
[tex]F(x) = 1 - e^{(-λx)}[/tex]
Now, we can plug in the values:
[tex]P(X ≥ 6) = 1 - F(6) \\ P(X ≥ 6) = 1 - (1 - e^{(-0.25 \times 6)}) \\ P(X ≥ 6) = 1 - (1 - e^{(-1.5)}) \\ P(X ≥ 6) = e^{(-1.5)}[/tex]
Therefore, the probability that the appliance will work at least six years without requiring repairs is approximately 0.2231 or 22.31%.
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The center of a circle is at (−2, −7) and its radius is 6.
What is the equation of the circle?
Responses
(x+2)2+(y+7)2=3
open parenthesis x plus 2 close parenthesis squared plus open parenthesis y plus 7 close parenthesis squared equals 3
(x+2)2+(y+7)2=36
open parenthesis x plus 2 close parenthesis squared plus open parenthesis y plus 7 close parenthesis squared equals 36
(x−2)2+(y−7)2=3
open parenthesis x minus 2 close parenthesis squared plus open parenthesis y minus 7 close parenthesis squared equals 3
(x−2)2+(y−7)2=36
option B is correct: [tex](x+2)^2+(y+7)^2=36[/tex]
The general equation of the circle is given by:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
where,
(h, k) is the center of the circle and r is the radius of the circle.
As per the statement:
The center of a circle is at (−2, −7) and its radius is 6.
[tex]\implies (h, k) = (-2, -7)[/tex] and [tex]r = 6[/tex] units
Substitute these we have:
[tex](x-(-2))^2+(y-(-7))^2=6^2[/tex]
[tex]\implies(x+2)^2+(y+7)^2=36[/tex]
Therefore, the equation of circle is, [tex]\bold{(x+2)^2+(y+7)^2=36}[/tex]
if all these statistics were analyzed at the end of the season, the correlation between number of wins and each of the four baseball statistics would be an example of
The correlation between the number of wins and each of the four baseball statistics would be an example of bivariate correlation analysis, which measures the strength and direction of the relationship between two variables.
The correlation between number of wins and each of the four baseball statistics would be an example of a bivariate correlation, which measures the strength and direction of the linear relationship between two variables. In this case, the two variables would be the number of wins and each of the four baseball statistics. The correlation coefficient would provide a numerical value that represents the degree of association between the two variables
A positive correlation coefficient would indicate that as one variable increases, so does the other, while a negative correlation coefficient would indicate that as one variable increases, the other decreases.
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If all these statistics were analyzed at the end of the season, the correlation between the number of wins and each of
the four baseball statistics would be an example of bivariate analysis.
Bivariate analysis involves analyzing the relationship between two variables to determine if there is any correlation or
association between them.
In this case, the two variables are the number of wins and each of the four baseball statistics.
By conducting a bivariate analysis, you can identify if there is any significant relationship between these variables and
potentially draw conclusions about their impact on team performance.
If all these statistics were analyzed at the end of the season, the correlation between the number of wins and each of
the four baseball statistics would be an example of bivariate analysis.
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Find the volume of the cone. Use 3.14 for pi. Round your answer to the nearest tenths
place.
The volume of the cone is approximately 37.7 cubic units
What is volume?
A volume is simply defined as the amount of space occupied by any three-dimensional solid. These solids can be a cube, a cuboid, a cone, a cylinder, or a sphere. Different shapes have different volumes.
To find the volume of a cone, we use the formula:
V = (1/3) * π * r² * h
where π is the constant pi, r is the radius of the base of the cone, and h is the height of the cone.
Plugging in the given values, we get:
V = (1/3) * 3.14 * 3² * 4 ≈ 37.7
Therefore, the volume of the cone is approximately 37.7 cubic units (rounded to the nearest tenth).
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now, using the above definition, determine if the function below is increasing, decreasing, even, odd, and/or invertible on its natural domain: $$f(x)
f'(x) = 2x - 2 is always positive on (-∞, 1) and always negative on (1, ∞), the function is increasing and decreasing, respectively, and therefore one-to-one. Therefore, f(x) is invertible in its natural domain.
What is the exponential function?
An exponential function is a mathematical function of the form f(x) = aˣ
where "a" is a constant called the base, and "x" is a variable. Exponential functions can be defined for any base "a", but the most common base is the mathematical constant "e" (approximately 2.71828), known as the natural exponential function.
To determine if the function f(x) = x² - 2x + 3 is increasing, decreasing, even, odd, and/or invertible in its natural domain, we need to analyze its derivative and second derivative:
f'(x) = 2x - 2
f''(x) = 2
Increasing/decreasing: Since f''(x) is always positive (i.e., 2 is positive), the function is always concave up and has a minimum at x = 1. Thus, f(x) is increasing on (-∞, 1) and decreasing on (1, ∞).
Even/odd: To determine if f(x) is even or odd, we need to check if it satisfies the properties of even and odd functions.
Even function: A function f(x) is even if f(-x) = f(x) for all x in the domain. Let's check if f(x) satisfies this property:
f(-x) = (-x)² - 2(-x) + 3 = x² + 2x + 3
f(x) = x² - 2x + 3
f(-x) ≠ f(x), so the function is not even.
Odd function: A function f(x) is odd if f(-x) = -f(x) for all x in the domain. Let's check if f(x) satisfies this property:
f(-x) = (-x)² - 2(-x) + 3 = x² + 2x + 3
-f(x) = -(x² - 2x + 3) = -x² + 2x - 3
f(-x) ≠ -f(x), so the function is not odd.
Invertible: To determine if f(x) is invertible, we need to check if it has an inverse function.
A function has an inverse function if it is one-to-one, which means that it passes the horizontal line test.
To check if f(x) is one-to-one, we can analyze its derivative.
Hence, f'(x) = 2x - 2 is always positive on (-∞, 1) and always negative on (1, ∞), the function is increasing and decreasing, respectively, and therefore one-to-one. Therefore, f(x) is invertible in its natural domain.
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Complete question:
For the following, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain: f(x) = x² - 2x + 3.
a company advertises that food preparation time can be significantly reduced with the handy dandy slicer. a sample of 12 individuals prepared the ingredients for a meal with and without the slicer. you are given the preparation times below. preparation times person with slicer without slicer 1 20 22 2 12 18 3 20 18 4 14 22 5 19 19 6 20 21 7 19 18 8 15 12 9 22 18 10 19 25 11 21 26 12 23 20 the null hypothesis should . a. be revised b. not be rejected c. be rejected
For the food preparation time data with or without slicer, the null hypothesis for this sample data should not be rejected. So, option (b) is right answer.
We have a food preparation time data of a company. The time of preparation is significantly reduced with the handy dandy slicer. Sample size, n = 12
We have a table present in above figure which has the preparation times data consists people, with slicer and without slicer. First we calculate difference between with slicer and without slicer data, that d: -2, -6, 2, -8 , 0, -1, 1, 3, 4, -6, -5, 3.
We do all work on the difference, d.
Mean value of difference d, [tex]\bar d = \frac{\sum_{i = 1}^{n} {d_i }}{n}[/tex] = -1.25
Standard deviations, [tex] \sigma = \sqrt{\frac{ \sum {d_i^2 - (\sum d_i)²}}{n - 1}}
[/tex] = 4.115
degree of freedom= 12 - 1 = 11
Let level of significance = 0.05
The null and alternative hypothesis are defined as [tex]H_0 : \mu_d = 0 [/tex]
[tex]H_a: \mu_d < 0[/tex]
Using t-test, [tex]t = \frac{ \bar d} {\frac{ \sigma}{ \sqrt{n}}}[/tex]
[tex]= \frac{ -1.25} {\frac{ 4.115}{ √12}}[/tex]
= - 1.05
Now, using t-distribution table, value of p for t = -1.05 for 11 degree of freedom is -1.796. So, p-value = -1.796 < 0.005, Hence, we should not reject the null hypothesis.
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Margo missed 24.6% of her free throw shots in a season. During the season, she shot a total of 90 free throws. Which of the following is the best estimate of the number of free throws Margo missed?
Answer: 22
Step-by-step explanation:
So since Margo missed 24.6% of 90 free throws, you have to find 24.6% of 90. The easiest way to do this is to do 90 multiplied by 0.246.
To multiply by percentages in general, move the decimal point to the left by two places. For example, if it said 82.6, the decimal point would go to the left two places and be 0.826.
Back to the original question...
so 90 multiplied by 0.246 is 22.14. If you can use a calculator, this is easy, if not just do the usual multiplication.
Then, this rounds to 22, and the problem is complete!
ruby is using the quadratic formula to solve a quadratic equation. which of the following is the next step for simplifying x=−6±85√2 responses
x=−3±8√5
x=−3±√5
x=−3±4√5
this is fully simplified.
4. Find volume. Show work, round to 2 decimal places.
*cone*
Answer: 80
Step-by-step explanation:
using definition of derivative:lim as h approaches 0f(a+h) - f(a) / h
Using the definition of the derivative, we have:
f'(a) = lim (h -> 0) [(f(a+h) - f(a)) / h]
We're asked to use the definition of the derivative, which includes the terms:
limit, h, f(a+h), f(a), and the fraction f(a+h) - f(a) / h.
The definition of the derivative of a function f(x) at a specific point x=a is given by:
f'(a) = lim (h -> 0) [(f(a+h) - f(a)) / h]
f(a+h) represents the value of the function f(x) at the point (a+h).
f(a) represents the value of the function f(x) at the point a.
f(a+h) - f(a) calculates the difference in the function values at these two points.
h is the small change in the x-coordinate (x-axis) between these two points, and we let h approach 0 to ensure we're finding the instantaneous rate of change.
(f(a+h) - f(a)) / h represents the average rate of change between the points (a, f(a)) and (a+h, f(a+h)).
By taking the limit as h approaches 0, we find the instantaneous rate of change at the point x=a, which is the derivative of the function f(x) at x=a.
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