a sphere has a radius of 6 feet. what are the dimensions of the cylinder with the greatest volume that can be inscribed in the sphere? what is the maximum volume of this cylinder?

The calculated value of the length of EA in the diagram is 16.7 cm

The diagram is an illustration of similar triangles, and the length of EA can be calculated using the following proportional equation

EA/EC = BD/DC

Where

EC = 9 + 6 = 15 cm

BD = 10 cm

DC = 9 cm

Substitute the known values in the above equation, so, we have the following representation

EA/15 = 10/9

Multiply both sides of the equation by 15

This gives

EA = 15 * 10/9

Evaluate the equation

EA = 16.7

Hence, the length of EA in the diagram is 16.7 cm

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i. The local maxima and minima are 3 and 2

ii. The absolute maximum of f(x,y) over the region is 3/2  at (1/√2, 1/√2), and the absolute minimum is -1/2, which is attained at (-1/√2, -1/√2).

iii.  There are no other critical points inside the disk, so we cannot tell whether they are extreme or saddle points.

i. To find the maxima and minima of the function f(x,y) = x² + y² + xy over the region {(x,y): x² + y² ≤ 1}, we first find the critical points by setting the partial derivatives equal to zero:

f(x) = 2x + y = 0

fy = 2y + x = 0

Solving these equations simultaneously gives the critical point (-1/3, 2/3). We now need to check if this is a local maximum, local minimum or a saddle point. To do this, we use the second partial derivative test.

f(xx) = 2, f(xy) = 1, fyy = 2

The determinant of the Hessian matrix is Δ = f(xx)f(yy_ - (fxy)² = 2(2) - (1)² = 3, which is positive, and f(xx) = 2, which is positive. Therefore, the critical point is a local minimum.

ii. To find the absolute maximum and minimum, we need to consider the boundary of the region. Let g(x,y) = x² + y² be the equation of the circle with radius 1 centered at the origin. We can parameterize this curve as x = cos(t) and y = sin(t), where 0 ≤ t ≤ 2π.

Substituting this into the function f(x,y), we get:

h(t) = f(cos(t), sin(t)) = cos²(t) + sin²(t) + cos(t)sin(t) = 1 + (1/2)sin(2t)

We now find the critical points of h(t) by setting dh/dt = 0:

dh/dt = cos(2t) = 0

This gives t = π/4 and 5π/4.

Substituting these values into h(t), we get:

h(π/4) = 3/2

h(5π/4) = -1/2

Therefore, the absolute maximum of f(x,y) over the region is 3/2, which is attained at (1/√2, 1/√2), and the absolute minimum is -1/2, which is attained at (-1/√2, -1/√2).

iii. There are no other critical points inside the disk, so we cannot tell whether they are extreme or saddle points.

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To answer your question, we first need to calculate the z-score formula. The z-score formula is:

z = (x - μ) / σ

Where:
x = the value we want to find the z-score for
μ = the mean of the distribution
σ = the standard deviation of the distribution

In this case, we are given that the distribution has a mean of 145.2 and a standard deviation of 16.8. We also know that we want to find the z-score for some performance values.

Let's say that the performance value we are interested in is 160. Using the z-score formula, we can calculate the z-score as:

z = (160 - 145.2) / 16.8
z = 0.88095

So the z-score for a performance value of 160 in this distribution is 0.88095.

It's worth noting that if the distribution is exactly normal, we can use a z-score table to find the percentage of values that fall below or above a certain z-score. However, if the distribution deviates from normality in any way, the z-score may not accurately represent the percentage of values in the distribution.
To calculate the z-score for a specific performance value in a distribution, you can use the following formula:

z-score = (value - mean) / standard deviation

Given the distribution has a mean of 145.2 and a standard deviation of 16.8, let's assume we have a specific performance value "X." You would then plug the numbers into the formula:

z-score = (X - 145.2) / 16.8

Replace "X" with the specific performance value you want to find the z-score for, and you'll have your answer.

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Answer: Using trigonometry, we can find the length of side AC. Since we know the length of side BC and one angle, we can use the tangent function:

tan(70) = AC/6

Multiplying both sides by 6, we get:

AC = 6 * tan(70)

Using a calculator, we get:

AC ≈ 19.22

Rounding to the nearest hundredth, we get:

AC ≈ 19.22 units.

Answer:2.33

Step-by-step explanation:

The Mean Value Theorem is a crucial theorem of calculus that reveals a relationship between the gradient of a curve and the values of its associated function at the endpoint.

Specifically, it states that provided f(x) is steady on the enclosed interval [a, b], and differentiable on (a, b), then there must exist a point c within the range of (a, b) such that

f(b) - f(a) = f'(c) * (b - a)

which translates to there being an individual c inside the parameterized region (a, b), such that the inclined angle of the tangent line to the graph at c is equal to the general incline of the graph between a and b.

The Mean Value Theorem possesses a plethora of utilities in mathematical analysis and calculus alike.

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The interaction degrees of freedom for this study is 8. This value indicates the number of independent comparisons that can be made to evaluate the interaction effect between academic status and class time on class attendance.

The interaction degrees of freedom in this scenario would be (5-1) x (3-1) = 8. This is because there are 5 levels of academic status (freshman, sophomore, junior, senior, and graduate students) and 3 levels of class time (8:00 a.m., 9:30 a.m., and 11:00 a.m.), resulting in 15 cells. However, the means for each cell have already been provided, which means that there is no need to calculate the main effects or the grand mean. Therefore, the degrees of freedom for the interaction effect can be calculated as (number of levels for academic status - 1) x (number of levels for class time - 1), which gives us 4 x 2 = 8. This represents the number of independent pieces of information that can be used to estimate the interaction effect between academic status and class time on class attendance.
The interaction degrees of freedom in a two-way ANOVA can be calculated using the formula: (r - 1) * (c - 1), where r is the number of rows (academic statuses) and c is the number of columns (class times). In this case, there are 5 academic statuses (freshman, sophomore, junior, senior, and graduate students) and 3 class times (8:00 a.m., 9:30 a.m., and 11:00 a.m.). Using the formula, the interaction degrees of freedom would be calculated as follows: (5 - 1) * (3 - 1) = 4 * 2 = 8.

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The calculated value of the slope of the line is -7

From the question, we have the following parameters that can be used in our computation:

The slope-intercept equation of a line is y = -7x - 2

This means that

y = -7x - 2

A linear equation is represented as

y = mx + c

Where

Slope = m

using the above as a guide, we have the following:

m = -7

This means that the slope of the line is -7

Hence, the slope of the line is -7

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The expected value for this binomial distribution is 0.5.

To find the expected value for the binomial distribution, we can use the formula:

E(X) = np

where:

X is the random variable representing the number of successes

n is the total number of trials

p is the probability of success in each trial

In this case, the binomial distribution has the following probabilities for the number of successes:

P(X=0) = 243/3125

P(X=1) = 162/625

P(X=2) = 216/625

P(X=3) = 48/625

P(X=4) = 32/3125

The total number of trials is the sum of the probabilities:

n = (243/3125) + (162/625) + (216/625) + (48/625) + (32/3125) = 1

The probability of success in each trial is the sum of the probabilities for X=1, X=2, X=3, and X=4:

p = (162/625) + (216/625) + (48/625) + (32/3125) = 0.5

Now we can use the formula to find the expected value:

E(X) = np = 1 * 0.5 = 0.5.

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Answer:

0.5616

Step-by-step explanation:

The expected value for a binomial distribution can be calculated using the formula E(X) = np, where n is the number of trials and p is the probability of success in each trial.

To calculate the expected value for the given binomial distribution, we need to multiply each number of successes by its corresponding probability and then sum them up.

0 successes: (0)(243/3125)

1 success: (1)(162/625)

2 successes: (2)(216/625)

3 successes: (3)(48/625)

4 successes: (4)(32/3125)

5 successes: (5)(1/3125)

Now, let's calculate each of these values:

0 successes: 0

1 success: 162/625

2 successes: 432/625

3 successes: 144/625

4 successes: 128/3125

5 successes: 5/3125

To find the expected value, we need to sum up these values:

0 + 162/625 + 432/625 + 144/625 + 128/3125 + 5/3125 = 0.5616

Therefore, the expected value for the given binomial distribution is approximately 0.5616.

To find the probability of picking 3 queens from the stack, we need to first find the total number of ways to pick 3 cards from the stack of 12. This is represented by the combination formula:

nCr = n! / (r! * (n-r)!)
where n is the total number of cards in the stack (12) and r is the number of cards we want to pick (3).
nCr = 12! / (3! * (12-3)!) = 220
So, there are 220 possible ways to pick 3 cards from the stack.
Now, we need to find the number of ways to pick 3 queens from the stack. Since there are 4 queens in the stack, we can use the combination formula again:
nCr = n! / (r! * (n-r)!)

where n is the number of queens in the stack (4) and r is the number of queens we want to pick (3).
nCr = 4! / (3! * (4-3)!) = 4
So, there are 4 possible ways to pick 3 queens from the stack.
Finally, we can find the probability of picking 3 queens by dividing the number of ways to pick 3 queens by the total number of ways to pick 3 cards:

P(3 queens) = 4 / 220 = 0.018 or approximately 1.8%.
To answer your question, let's calculate the probability of picking 3 queens from the stack of 12 cards containing 4 aces, 4 kings, and 4 queens.
The total number of ways to pick 3 cards from the stack of 12 cards is represented by the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of cards and k is the number of cards chosen. In this case, n=12 and k=3.

C(12, 3) = 12! / (3!(12-3)!) = 12! / (3!9!) = (12 × 11 × 10) / (3 × 2 × 1) = 220

calculate the number of ways to pick 3 queens from the 4 queens available:
C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 × 3 × 2) / (3 × 2 × 1) = 4
Finally, divide the number of ways to pick 3 queens by the total number of ways to pick 3 cards to find the probability:
Probability = (Number of ways to pick 3 queens) / (Total number of ways to pick 3 cards) = 4 / 220 = 1/55 ≈ 0.0182
So, the probability of picking 3 queens from the stack is approximately 0.0182 or 1/55.

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The Schwarz Lemma is a result in complex analysis that gives information about analytic functions that map the open unit disc to itself and have a fixed point at the origin.

The first part of the theorem states that if f is analytic on the open unit disc D and f(0) = 0, then |f(z)| ≤ |z| for all z in D, and |f'(0)| ≤ 1. This means that the absolute value of f(z) is always less than or equal to the absolute value of z, and the absolute value of the derivative of f at the origin is less than or equal to 1.

The second part of the theorem states that if there exists a point zo in D{0} such that either |f(zo)| = |zo| or f'(0) = 1, then f must be a rotation of the disc, i.e., f(z) = cz for some complex number c with |c| = 1. In particular, if f(z0) = z0 or f'(0) = 1, then c = 1 and f = id, the identity function.

The Schwarz Lemma is an important tool in complex analysis for studying functions that preserve the unit disc, and has applications in areas such as conformal mapping and geometric function theory.

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We can say with 75.8% confidence that the true mean speed of all cars on this section of highway falls within the range of 65.181 to 66.819 mph.

Based on the information provided, we can use the formula for a confidence interval for a population mean:
CI = X ± z*(σ/√n)
Where:
CI = Confidence interval
X = Sample mean (66 mph)
z = z-score for the desired confidence level (75.8% = 0.758, so we can find the corresponding z-score using a standard normal distribution table or calculator. For a one-tailed test, the z-score is approximately 0.69)
σ = Standard deviation of the population (9.5 mph)
n = Sample size (64)
Plugging in the values, we get:
CI = 66 ± 0.69*(9.5/√64)
CI = 66 ± 0.69*(1.1875)
CI = 66 ± 0.8194
Rounding to 3 decimal places, we get:
CI = (65.181, 66.819)
Therefore, we can say with 75.8% confidence that the true mean speed of all cars on this section of highway falls within the range of 65.181 to 66.819 mph.

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The Height of the Cone is 6 cm.

The shape's volume is equal to the product of its area and height. = Height x Base Area = Volume.

The formula for the volume of a cone is V=1/3hπr².

Volume of Cone= 1/3 πr²h

where r is the radius and h is the height.

Now, if V= 32π cm³ and r= 4 cm

Then, Volume of Cone = 1/3 πr²h

                            32π = 1/3 π(4)²h

                             32 = 1/3  (4)²h      

                               32= 1/3 (16)h

                               h/3 = 2

                              h= 6cm

Hence, the height is 6 cm.

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The indefinite integral of (1 / 1+e^12x) is (1/12) ln|1+e^12x| + C, where C is the constant of integration.

To find the indefinite integral of (1 / 1+e^12x), we can use a table of integrals with forms involving eu. The form that matches our integral is ∫(1 / 1+e^u) du, where u=12x.

We can substitute u=12x and du/dx=12 to get ∫(1 / 1+e^12x) dx = (1/12) ∫(1 / 1+e^u) du.

Using the table of integrals, the integral of (1 / 1+e^u) du is ln|1+e^u| + C, where C is the constant of integration.

Substituting back in u=12x and multiplying by 1/12, we get the final answer: ∫(1 / 1+e^12x) dx = (1/12) ln|1+e^12x| + C.

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The  integral of the function √x, y=2 and the y  is   G. 8/3

You want the area between y=2 and y=√x.

Bounds

The square root curve is only defined for x ≥ 0. It will have a value of 2 or less for m √x ≤ 2

x ≤ 4 . . . . square both sides

So, the integral has bounds of 0 and 4.

Integral

The integral is

[tex]\int\limits^4_0 {[2-xx^{1/2} } \, dx = \int\limits^4_6 {2x-2/3x^{4/3} } \, dx = 8-2/3(\sqrt{4)x^{3} } =8/3[/tex]

Additional comment

You will notice that this is 1/3 of the area of the rectangle that is 4 units wide and 2 units high. That means the area inside a parabola is 2/3 of the area of the enclosing rectangle. This is a useful relation to keep in the back of your mind.

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From the tree diagram, the family have 2 × 3 × 5 = 30 choices.

Here, the types of cars to be considered are minivan or an SUV.

Those vehicles come in red, gold, green, silver, and blue.

And each vehicle has three models i.e., standard, sport, or luxury.

First we draw the tree diagram.

The required tree diagram for this siuation is shown below.

Since for each type of vechicle  has three models, the number of choices for two vehicles would be,

2 × 3 = 6

And these vehicles come in red, gold, green, silver, and blue.

So, the number of choices the family have:

6 × 5

i.e., 2(types of cars) × 3(types of models of each vehicle) × 5(colors in each model)

so, the family have 2 × 3 × 5 = 30 choices.

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a. This is a very low probability, indicating that the new system implemented by Kroger is effective in reducing waiting times.

b. The probability that a customer will have to wait between 2 and 4 seconds is approximately 0.5433.

c. The probability that a customer will have to wait more than 5 minutes (300 seconds) is approximately 0.000006, or 0.0006%.

a. The probability density function of waiting time applicable at Kroger is the exponential distribution function.

b. The probability of a customer having to wait between 2 and 4 seconds can be calculated as follows:

Let λ be the rate parameter of the exponential distribution, which represents the average number of customers served per second. Since the waiting times are exponentially distributed, the probability density function of the waiting time t is given by:

[tex]f(t) = \lambda \times e^{(-\lambda\times t)}[/tex]

We want to find the probability that a customer will have to wait between 2 and 4 seconds. This can be calculated as the difference between the cumulative distribution functions (CDF) evaluated at 4 seconds and 2 seconds:

P(2 < t < 4) = F(4) - F(2)

where F(t) is the CDF of the exponential distribution:

[tex]F(t) = 1 - e^{(-\lambda \times t)}[/tex]

Substituting the value of λ (which we need to estimate), we can solve for the probability:

[tex]P(2 < t < 4) = (1 - e^{(-\lambda4)}) - (1 - e^{(-\lambda2)})\\= e^{(-\lambda2)} - e^{(-\lambda4)}[/tex]

To estimate λ, we can use the information given in the problem that the average waiting time is "just seconds". Let's assume that this means an average waiting time of 2 seconds. Then, the rate parameter λ can be estimated as:

λ = 1 / 2

Substituting this value in the equation above, we get:

[tex]P(2 < t < 4) = e^{(-1)} - e^{(-2)[/tex]

≈ 0.5433

c. The probability of a customer having to wait more than 5 minutes (i.e., 300 seconds) can be calculated as follows:

P(t > 300) = 1 - F(300)

where F(t) is the CDF of the exponential distribution as given above. Substituting the value of λ estimated earlier, we get:

[tex]P(t > 300) = 1 - (1 - e^{(-\lambda300)})\\= e^(-\lambda300)[/tex]

Substituting the value of λ, we get:

[tex]P(t > 300) = e^{(-150)}[/tex]

≈ 0.000006

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The solutions are given below,

(a) f(g(x)) = 2x² - 4x + 2

(b) g(f(x)) = 2x² - 1

(c) f(f(x)) = 8x⁴

To find f(g(x)), we substitute g(x) into the function f(x):

f(g(x)) = 2(g(x))²

f(g(x)) = 2(x-1)²

f(g(x)) = 2(x² - 2x + 1)

f(g(x)) = 2x² - 4x + 2

Therefore, f(g(x)) = 2x² - 4x + 2.

b. To find g(f(x)), we substitute f(x) into the function g(x):

g(f(x)) = f(x) - 1

g(f(x)) = 2x² - 1

Therefore, g(f(x)) = 2x² - 1.

c. To find f(f(x)), we substitute f(x) into the function f(x):

f(f(x)) = 2(f(x))²

f(f(x)) = 2(2x²)²

f(f(x)) = 2(4x⁴)

f(f(x)) = 8x⁴

Therefore, f(f(x)) = 8x⁴.

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By Stoke's theorem, the flux of the vector field F across surface S is equal to the line integral of F over the boundary curve C: Flux = ∮C (F ⋅ dr) = 20

To find the flux of the vector field F = 3i + 2j + 4k across the surface S using Stoke's theorem, we first need to find the curl of F: Curl(F) = (∂Fz/∂y - ∂Fy/∂z)i - (∂Fx/∂z - ∂Fz/∂x)j + (∂Fy/∂x - ∂Fx/∂y)k Since Fz = 4, Fy = 2, and Fx = 3, all their partial derivatives are constants: Curl(F) = (0)i - (0)j + (0)k = 0

Now, let's find the line integral over the boundary curve C: ∮C (F ⋅ dr) = ∫₀^4 3dx + ∫₀^2 2dy + ∫₀^1 4dz We can integrate each part separately: ∫₀^4 3dx = 3(4) - 3(0) = 12 ∫₀^2 2dy = 2(2) - 2(0) = 4 ∫₀^1 4dz = 4(1) - 4(0) = 4

Now, add up the results: ∮C (F ⋅ dr) = 12 + 4 + 4 = 20

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The surface area of the container is 314 sq ft

From the question, we have the following parameters that can be used in our computation:

length of 5 ft, a width of9 ft, and a height of 8 ft.

The surface area is calculated as

Area = 2 * (Length * Width + Length * Height + Width * Height)

Substitute the known values in the above equation, so, we have the following representation

Area = 2 *(5 * 9 + 5 * 8 + 9 * 8)

Evaluate

Area = 314

Hence, the area is 314 sq ft

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To calculate the present value of the winnings, we would need to know the total amount of the winnings and the interest rate or discount rate used to calculate the present value. Without that information, it is impossible to provide an accurate answer.

Please provide more information for a precise response. To answer your question, I need to know the details of the winnings, such as the amount, number of payments, and the interest rate. Without this information, I cannot calculate the present value. However, I can explain the steps to calculate the present value of the winnings when the first payment comes immediately.

1. Determine the amount of each payment (winnings).
2. Identify the number of payments.
3. Identify the interest rate used for discounting future payments.
4. Calculate the present value of each payment using the formula:

PV = Payment / (1 + interest rate) ^ n

where PV is the present value, n is the number of periods (e.g., years) into the future that the payment occurs, and the interest rate is in decimal form (e.g., 5% = 0.05).

5. Add the present values of all payments together to get the total present value of the winnings. Remember not to round any intermediate calculations and to enter your final answer in millions rounded to 2 decimal places.

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Answer:

There can be no answer, as you did not provide the box plot to compare the data.

This function gives us the perimeter in meters for any given length x of the rectangle. To express the given quantity as a function f(x) of one variable x, we need to use the given information about the area and the formula for the perimeter of a rectangle.

Let's start by recalling the formula for the area of a rectangle:

A = length x width

We know that the area of the rectangle is 187 square meters, so we can write:

187 = x y

Now, let's recall the formula for the perimeter of a rectangle:

P = 2(length + width)

We want to express the perimeter as a function of x only, so we need to eliminate y from this formula using the information we have about the area:

y = 187/x

Substituting this expression for y into the formula for the perimeter, we get:

P = 2(x + 187/x)

Therefore, the function f(x) that expresses the perimeter of the rectangle as a function of its length x is:

f(x) = 2(x + 187/x)

This function gives us the perimeter in meters for any given length x of the rectangle.

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Answer:

12/40

Step-by-step explanation:

to do multiplication with fractions is super simple you just have to multiply the numerators and denominators so the 2 top numbers (4 x 3) and the two bottom numbers (5 x 8) and create your fraction (12/40)

Answer:3/10

Step-by-step explanation: you first multiply the 2 numerator 4*3=12

Then multiply the 2 denominators 5*8=40 now you have 12/40 to simplify you divide both by 4 so you have  3/10

4.543, [tex]4\frac{11}{20}[/tex],4.57, 37/8 is the order from least to greatest

The given numbers are 4.543, 4.57, [tex]4\frac{11}{20}[/tex], 37/8

We have to order from least to greatest

Let us find the decimal values which are given in fraction form

[tex]4\frac{11}{20}[/tex] = 91/20

= 4.55

Now let us find 37/8 in decimal form'

37/8 = 4.625

Now 4.543, 4.57, 4.55, 4.625 arrange from least to greatest

By comparing the decimal values we get an order from least to greatest is 4.543, 4.55, 4.57, 4.625

Hence, 4.543, [tex]4\frac{11}{20}[/tex],4.57, 37/8 is the order from least to greatest

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The average speed for the remaining part of his journey is A = 280 km/hr

Given data ,

We are given that the man drove 360 km from Singapore to Kuala Lumpur in 4 1/2 hours, which is equivalent to 4.5 * 60 = 270 minutes.

Therefore, his overall average speed is:

average speed = 360 km / 270 min

= 1.333... km/min

We must know the distance and time he covered during that portion of the voyage in order to calculate his average speed for the remaining distance. We know he traveled the following distance in two hours at an average speed of 110 km/h on a highway:

distance = speed x time = 110 km/hr × 2 hr = 220 km

Therefore, the distance he traveled during the remaining part of the journey is:

The distance = total distance - distance on highway = 360 km - 220 km = 140 km

Additionally, we know that he used 37.5 litres of petrol for the entire trip. Assume his car uses the same amount of fuel throughout the entire trip. His fuel efficiency may therefore be computed as follows:

fuel efficiency = total distance / petrol used

F = 360 km / 37.5 litres

F = 9.6 km/litre

We can use this fuel efficiency to calculate the time he spent on the remaining part of the journey, since time = distance / speed and speed = distance / petrol used:

Time = distance / speed

T = 140 km / (fuel efficiency × petrol used)

T = 140 km / (9.6 km/litre × 37.5 litres)

T = 0.5 hours

Therefore, his average speed for the remaining part of the journey is:

Now , the average speed = distance / time

A = 140 km / 0.5 hours

A = 280 km/hr

Hence , his average speed for the remaining part of the journey was 280 km/hr

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The problem asks for the number of different samples of size 2 that can be selected from a population of size 10. To solve this problem, we can use the formula for the number of combinations of n objects taken r at a time, which is given by nCr = n!/(r!(n-r)!), where n is the size of the population and r is the size of the sample.

In this case, we have n=10 and r=2, so the number of different samples of size 2 that can be selected from a population of size 10 is given by 10C2 = 10!/(2!(10-2)!) = 45. Therefore, there are 45 different samples of size 2 that can be selected from a population of size 10.

Another way to think about this problem is to consider that when selecting a sample of size 2 from a population of size 10, we can choose the first element from any of the 10 objects in the population, and then choose the second element from the remaining 9 objects in the population (since we can't choose the same object twice).

Therefore, the total number of different samples of size 2 that can be selected is 10 x 9 = 90. However, since the order in which we choose the elements of the sample doesn't matter, we need to divide by 2 (the number of ways to arrange 2 elements), giving us a total of 45 different samples of size 2.

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Complete question:

How many different samples of size 2 can be selected from a population of size 10?

As we can observe in the attachment below figure the ∠b, ∠g, and ∠f are the right angles.

We must ascertain whether an angle is exactly 90 degrees in order to decide whether it is a right angle or not. There are several methods for doing this:

Therefore, The angles at ∠b, ∠g, and ∠f are right angles, as shown in the attachment below.

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To answer your question, we need to calculate the test statistic for the hypothesis.

Based on the information provided, we have:

- Number of total responses (n) = 830
- Number of positive responses (x) = 439

Assuming you want to test the proportion of positive responses, we can use the formula for the test statistic in a one-sample proportion hypothesis test:

z = (p_hat - p0) / sqrt(p0(1-p0)/n)

where p_hat is the sample proportion, p0 is the null hypothesis proportion, and n is the total number of responses. First, let's calculate p_hat:

p_hat = x/n = 439/830 ≈ 0.529

Now, to determine the test statistic, we need to know the null hypothesis proportion (p0). If you provide that information, I can help you calculate the test statistic (z).

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