Need help with this probability worksheet

(1) The simplest polynomial function with the zeros of 2, -1, and 1 is x³ - 2x² - x + 2.

(2) The simplest polynomial function with the zeros of  0, -2, and √3 is x³ + 2√3x² - 2x² - 2√3x.

(3) The simplest polynomial function with the zeros of  2i, 1 and -2 is x³ + 2x² - 5x - 8.

The simplest polynomial function with the zeros of 2, -1, and 1 is determined as;

f(x) = (x - 2)(x + 1)(x - 1)

f(x) = (x² - x - 2)(x - 1)

f(x) = x³ - 2x² - x + 2

The simplest polynomial function with the zeros of  0, -2, and √3 is calculated as;

f(x) = x(x + 2)(x - √3)

f(x) = x³ + 2√3x² - 2x² - 2√3x

The simplest polynomial function with the zeros of  2i, 1 and -2 is calculated as;

f(x) = (x - 2i)(x + 2i)(x - 1)(x + 2)

f(x) = (x² + 4)(x - 1)(x + 2)

f(x) = x³ + 2x² - 5x - 8

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The tangent line has the largest slope at x = √(32/5), with a slope of approximately 254.86.

To find the point on the line 3x + 5y + 5 = 0 which is closest to the point (5,2), we need to use the formula for the distance between a point and a line. The formula is:

distance = |ax + by + c| / √(a² + b²)

where (a,b) is the direction vector of the line and (x,y) is any point on the line. In this case, the direction vector is (3,5) and we can find a point on the line by setting y = 0:

3x + 5(0) + 5 = 0
x = -5/3

So the point on the line closest to (5,2) is:

distance = |3(-5/3) + 5(0) + 5| / √(3² + 5²) = 4 / √34

To find the value(s) of x on the curve y = -7 + 160x³ - 3x^5 where the tangent line has the largest slope, we need to find the derivative of y with respect to x and set it equal to zero:

y' = 480x² - 15x^4
480x² - 15x^4 = 0
x²(32 - 5x²) = 0
x = 0 or x = ±√(32/5)

We can now find the slope of the tangent line at each of these values of x:

slope at x = 0: y' = 0, so the tangent line is horizontal and has slope 0
slope at x = √(32/5): y' = 480(32/5) - 15(32/5)³ ≈ 254.86
slope at x = -√(32/5): y' = 480(32/5) - 15(-32/5)³ ≈ -254.86

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The surface area of the right rectangular prism with given dimensions is equal to 207.4 square centimeters.

Surface area of the right rectangular prism

= 2 ( length × width + width × height + height × length )

In the diagram,

length of the right rectangular prism = 6.7cm

Width of the right rectangular prism = 6.0cm

Height of the right rectangular prism = 5cm

Substitute the values in the formula we get,

Surface area of the right rectangular prism

= 2 × ( 6.7 × 6 + 6 × 5 + 5 × 6.7 )

= 2 × ( 40.2 + 30 + 33.5 )

= 2 ×  103.7

= 207.4 square centimeters.

Therefore, the surface area of the prism with length 6.7 cm , width 6.0 cm and height 5cm is equal to 207.4 square centimeters.

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Answer:

Dividing this figure into rectangles, and then adding the areas of those rectangles:

4(24) + 4(12) + 8(8) + 16(16)

= 96 + 48 + 64 + 256 = 464 square units

(1)  The samples being compared should be independent of each other.

(2) The populations from which the samples are drawn should follow a normal distribution.

(3) The two samples being compared should have equal variances.

To use a two-sample f-test, three conditions must be met.

Firstly, the samples being compared should be independent of each other. Independence means that the observations in one sample do not influence or depend on the observations in the other sample. This condition is important to ensure that the variability between the samples is not confounded or biased by any relationship or dependence between the observations.

Secondly, the populations from which the samples are drawn should follow a normal distribution. The assumption of normality is required for the f-test to accurately assess the differences in means between the two groups. If the data deviates significantly from a normal distribution, alternative non-parametric tests may be more appropriate.

Finally, the two samples being compared should have equal variances. This assumption, known as the assumption of equal variances or homogeneity of variances, implies that the variability within each group is similar. Violation of this assumption can affect the accuracy of the f-test results. In cases where the variances are unequal, modified versions of the f-test, such as the Welch's t-test, can be used.

In summary, the three conditions for using a two-sample f-test are independence between samples, normality of the populations, and equality of variances. These conditions ensure that the f-test accurately evaluates the differences in means between the two groups being compared.

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In the circle the measure of the angle is given as 31°

The measure of a particular arc can be computed by splitting its length (s) up, and then dividing it by the radius of the circle (r).

The outcome is measured in radians, but this can be converted to degrees too - just multiply it with 180 and divide it all by pi (3.14).

From the figure we have

25 + 37 = 62

62 / 2

= 31 degrees

Hence the calculated measure of Y is given as 31 degrees

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Yes, the set S of all polynomials of the form c1 + c2x + c3x^3, with c1, c2, and c3 belonging to the real numbers (R), is a vector space. To prove this, we need to show that S satisfies the following properties:

1. Closure under addition: Given any two polynomials P1 = a1 + a2x + a3x^3 and P2 = b1 + b2x + b3x^3 in S, their sum P1 + P2 = (a1 + b1) + (a2 + b2)x + (a3 + b3)x^3 is also in S, since the coefficients are real numbers.

2. Closure under scalar multiplication: Given a polynomial P = c1 + c2x + c3x^3 in S and a scalar k ∈ R, the product kP = (kc1) + (kc2)x + (kc3)x^3 is also in S, as the coefficients are still real numbers.

3. Existence of additive identity: The zero polynomial, 0 = 0 + 0x + 0x^3, is in S, and adding it to any polynomial P in S results in P itself (P + 0 = P).

4. Existence of additive inverse: Given a polynomial P = c1 + c2x + c3x^3 in S, its additive inverse -P = (-c1) + (-c2)x + (-c3)x^3 is also in S. Adding P and -P results in the zero polynomial (P + (-P) = 0).

5. Associativity and commutativity of addition, as well as distributive properties of scalar multiplication, are inherited from the real numbers.

Thus, S is a vector space since it satisfies all the required properties.

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If f(x)=[tex](2-10x)(2-10x^2)(2-10x^3)(2-10x^4)(2-10x^5)(2-10x^6)[/tex], then its derivative, f'(1)= 5/2.

To find the derivative of the given function f(x), we can use the product rule of differentiation. Let's start by taking the natural logarithm of f(x) to simplify the calculation:

ln[f(x)] = ln[[tex](2-10x)(2-10x^2)(2-10x^3)(2-10x^4)(2-10x^5)(2-10x^6)[/tex]]

= [tex]ln(2-10x) + ln(2-10x^2) + ln(2-10x^3) + ln(2-10x^4) + ln(2-10x^5) + ln(2-10x^6)[/tex]

Now, we can take the derivative of both sides using the chain rule and product rule of differentiation:

[d/dx] ln[f(x)] = [d/dx] [tex][ln(2-10x) + ln(2-10x^2) + ln(2-10x^3) + ln(2-10x^4) + ln(2-10x^5) + ln(2-10x^6)][/tex]

d[f(x)]/f(x) = [tex]1/(2-10x) + 1/(2-10x^2) + 1/(2-10x^3) + 1/(2-10x^4) + 1/(2-10x^5) + 1/(2-10x^6)[/tex]

Now, we can evaluate f'(1) by substituting x=1 into the above expression:

d[f(x)]/dx| x=1 = 1/(2-10) + 1/(2-10) + 1/(2-10) + 1/(2-10) + 1/(2-10) + 1/(2-10)

= -10/(-8) - 20/(-8) - 30/(-8) - 40/(-8) - 50/(-8) - 60/(-8)

= 5/2

Therefore, the value of f'(1) is 5/2.

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It is true that the When data collection involves online surveys, the process of data validation involves examining if instructions were followed precisely.

This is because online surveys often have specific instructions that respondents are expected to follow. Data validation ensures that the data collected is accurate, reliable, and valid. It involves checking for errors or inconsistencies in the data, as well as making sure that respondents have answered all questions correctly. This process helps to ensure that the data collected is of high quality and can be used for analysis and decision-making purposes. Additionally, data validation can also help to identify areas where improvements can be made to the survey design or data collection process.

Data validation in online surveys refers to ensuring the accuracy and quality of the collected data. It involves checking for inconsistencies, errors, and incomplete responses. While following instructions precisely is important, data validation focuses on data accuracy, preventing duplicate responses, and verifying if respondents meet the target demographic. It aims to improve the reliability of the data and reduce the margin of error, ensuring meaningful conclusions can be drawn from the results.

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Answer:

Step-by-step explanation:

finding the smallest possible distance from the line to the origin follows as

the normal vector=(-1,-2,1)

using direction vector we need to create the equation of the plane

-1(x-2)-2(y-3)+1(z+7)=0

we get;

-x+2-2y+6+z+7=0

-x-2y+z+13=0

x=2-t; y=2-3t; z=-7+t

on substituting;

-1(2-t)-2(3-2t)+1(-7+t)+13=0

-2+t-6+4t-7+t+13=0

we get;

t=1

so point is(1,1,-6)

distance from point to origin

d=[tex]\sqrt{(1^2+1^2+(-6)^2)}[/tex]=[tex]\sqrt{38}[/tex]

therefore

the answer is [tex]\sqrt{38}[/tex]=6.16

Using the formula P(A?B) = P(A) + P(B) - P(A and B), we can find the probability of both A and B:

P(A and B) = P(A) + P(B) - P(A?B) = 0.50 + 0.45 - 0.80 = 0.15

Using the formula P(A|B) = P(A and B) / P(B), we can find the conditional probability of A given B:

P(A|B) = P(A and B) / P(B) = 0.20 / 0.40 = 0.5

To check if A and B are independent, we need to see if P(A|B) = P(A).

P(A) = 0.30

P(A|B) = 0.50

Since P(A|B) is not equal to P(A), A and B are not independent.

The probability of at least one foul-up occurring can be found using the formula:

P(at least one foul-up) = 1 - P(no foul-up)

We can find P(no foul-up) by using the fact that A, B, and C are independent events:

P(no foul-up) = P(not A) * P(not B) * P(not C)

= (1 - 0.02) * (1 - 0.01) * (1 - 0.04)

= 0.9304

Therefore,

P(at least one foul-up) = 1 - P(no foul-up)

= 1 - 0.9304

= 0.0696

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Answer:

To find the mode, we look for the number that appears most frequently in the set:

The mode is 34 because it appears twice, while all other numbers appear only once.

To find the median, we need to arrange the numbers in numerical order:

13, 13, 26, 34, 34, 41, 53, 61, 69

The median is 34, which is the middle number when the set is arranged in order.

To find the mean, we add up all the numbers and divide by the total number of numbers:

(53 + 13 + 34 + 41 + 26 + 61 + 34 + 13 + 69) / 9 = 35.89 (rounded to two decimal places)

The mean is approximately 35.89.

To find the range, we subtract the smallest number from the largest number:

69 - 13 = 56

The range is 56.

Therefore, the mode is 34, the median is 34, the mean is approximately 35.89, and the range is 56.

Answer:

Step-by-step explanation:

mean is 35.89

the median is 34

the mode is 34

and the rang is 56

Answer:

A

Step-by-step explanation:

Continuous data is a continuous scale that covers a range of values without gaps, interruptions, or jumps.

The probability of a collision, if the deer needs 2 seconds to cross the road, is 0.181.

We have,

Since the traffic on Rosedale Road follows a Poisson process with a rate of 6 cars per minute, the number of cars passing in a given time period follows a Poisson distribution with a mean:

λ = (6 cars/min) x (1 min/60 s) = 0.1 cars per second.

To find the probability of a collision if the deer needs 5 seconds to cross the road, we can use the Poisson distribution to calculate the probability of at least one car passing in the next 5 seconds.

Let X be the number of cars passing in 5 seconds.

Then X follows a Poisson distribution with a mean of λ5 = 0.15 = 0.5.

So,

P(at least one car in 5 seconds)

= 1 - P(no cars in 5 seconds)

= 1 - e^(-0.5)

≈ 0.393

This is the probability of a collision if the deer needs 5 seconds to cross the road.

If the deer only needs 2 seconds to cross the road, then we need to calculate the probability of at least one car passing in the next 2 seconds.

Let Y be the number of cars passing in 2 seconds.

Then Y follows a Poisson distribution with a mean of λ2 = 0.12 = 0.2.

P(at least one car in 2 seconds)

= 1 - P(no cars in 2 seconds)

= 1 - e^(-0.2)

≈ 0.181

This is the probability of a collision if the deer needs 2 seconds to cross the road.

Thus,

The probability of a collision, if the deer needs 2 seconds to cross the road, is 0.181.

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The following are the measures for the vertical angles:

13). m∠SYX = 76°, (14). m∠XYW = 40°, (15). m∠WYV = 64°, (16). m∠SYW = 116°, (17). m∠TYX = 140°, and (18). m∠VYX = 104°

Vertical angles also called vertically opposite angles are formed when two lines intersect each other, the opposite angles formed by these lines are vertically opposite angles and are equal to each other.

We shall evaluate for the measure of the angles as follows:

13). m∠SYX = m∠UYV = 76°

(14). m∠XYW = m∠UTY = 40°

(15). m∠WYV = m∠SYT = 64°

(16). m∠SYW = m∠TYV = 76° + 40° = 116°

(17). m∠TYX = m∠UYW = 76° + 64° = 140°

(18). m∠VYX = m∠SYU = 64° + 40° = 104°

Therefore, the measures for the vertical angles are:

13). m∠SYX = 76°, (14). m∠XYW = 40°, (15). m∠WYV = 64°, (16). m∠SYW = 116°, (17). m∠TYX = 140°, and (18). m∠VYX = 104°

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The number of candidates (n) and the number to be selected (r) were not provided in the question, I am unable to provide a specific numerical answer. There are 252 ways to select a list of 5 candidates from a pool of 10 candidates.

If we assume that the number of candidates is n and the boss wants to select a list of k candidates, then the number of ways to select the list can be calculated using the combination formula.
The formula for combination is: n choose k = n!/k!(n-k)!
Using this formula, we can find the number of ways to select the list of k candidates from the n candidates.
For example, if there are 10 candidates and the boss wants to select a list of 5 candidates, then the number of ways to select the list would be:
10 choose 5 = 10!/5!(10-5)! = 252
Therefore, there are 252 ways to select a list of 5 candidates from a pool of 10 candidates.
In general, the number of ways to select a list of k candidates from a pool of n candidates can be calculated using the combination formula as n choose k = n!/k!(n-k)!.
Hi there! It seems that your question is incomplete, but I'll try my best to provide a helpful answer using the information provided. It appears you are asking about the number of ways to select a list of candidates to submit to a boss for a final decision.
To calculate the number of ways to select a list of candidates, we will use the concept of combinations. A combination is a way of selecting items from a larger set, without considering the order in which they are chosen. The formula for combinations is given by:
C(n, r) = n! / (r!(n-r)!)
where C(n, r) represents the number of combinations, n is the total number of candidates, r is the number of candidates to be selected, and ! denotes a factorial (e.g., 5! = 5 x 4 x 3 x 2 x 1).
However, If you can provide the values for n and r, I would be happy to help you calculate the number of ways to select the list of candidates to submit to the boss.

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The value of angle c is 86.

We have,

a = 68

b = 26

We are assuming that in the triangle there are three angles:

a, b, and c

Now,

The sum of the angles in the triangle is 180.

So,

a + b + c = 180

68 + 26 + c = 180

94 + c = 180

c = 180 - 94

c = 86

Thus,

The value of angle c is 86.

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The complete question.

Given that angle a = 68° and angle b = 26° in the triangle.

Find the angle c.

The sample variance of bottle weight is 2.80.The sample variance is a measure of how spread out the data is from the mean. In this case, the sample variance of 2.80 means that the bottle weights vary quite a bit from the sample mean of 4 grams.  

To find the sample variance, first we need to calculate the sample mean, which is (4+2+5+4+5+2+6)/7 = 4.

Then, we subtract the sample mean from each observation, and square each of the differences: (4-4)^2, (2-4)^2, (5-4)^2, (4-4)^2, (5-4)^2, (2-4)^2, and (6-4)^2.

The sum of these squared differences is 28. Finally, we divide this sum by n-1 (where n is the sample size) to get the sample variance: 28/6 = 2.80.

However, it's important to note that the sample variance is just an estimate of the true population variance, and can be affected by outliers or the specific sample chosen.

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According to unitary method, Marques made 2 three-point shots and 6 free throws in the game.

Let's say Marques made x three-point shots and y free throws. Since a three-point shot is worth three points and a free throw is worth one point, we can write two equations to represent the total number of points scored and the total number of shots made:

3x + y = 12 (equation 1)

x + y = 8 (equation 2)

We can now use the unitary method to solve for x and y. We want to find the value of one three-point shot, so we can rearrange equation 1 to get:

3x = 12 - y

x = (12 - y) / 3

Similarly, we want to find the value of one free throw, so we can rearrange equation 2 to get:

y = 8 - x

We can substitute the second equation into the first equation to get:

3x = 12 - (8 - x)

3x = 4 + x

2x = 4

x = 2

Now that we know that Marques made 2 three-point shots, we can substitute this value into equation 2 to get:

y = 8 - x

y = 8 - 2

y = 6

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The angular coordinate, angular velocity, and angular acceleration of the flywheel are: (a) At t = 0, θ = θ0 = 0.5 rad, ω = 0, and α = 12π²θ0 = 23.55 rad/s².

(b) At t = 0.125 s, θ = 0.267 rad, ω = 4.116 rad/s, and α = -69.08 rad/s².

The given equation for the angular displacement of the flywheel is θ=θ0e(-3πcos(4πt)). Here, θ0 = 0.5 rad. To find the angular velocity and angular acceleration, we need to differentiate θ with respect to time.

θ = θ0e(-3πcos(4πt))

ω = dθ/dt = -12π²θ0e(-3πcos(4πt))sin(4πt)

α = d²θ/dt² = -48π³θ0e(-3πcos(4πt))(cos(4πt) - 2)sin(4πt)

Substituting t = 0, we get:

(a) At t = 0, θ = θ0 = 0.5 rad, ω = dθ/dt = 0, and α = d²θ/dt² = 12π²θ0 = 23.55 rad/s².

(b) At t = 0.125 s, θ = 0.267 rad, ω = dθ/dt = 4.116 rad/s, and α = d²θ/dt² = -69.08 rad/s².

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The point which represents the number of dollars Carola makes per hour is (1, 8).

Two variables have a proportional relationship if all the ratios of the variables are equivalent.

The constant of proportionality is the ratio of the y value (total earnings) to the x value (hours worked). That is:

constant of proportionality (k) = y/x

Since Carola works for 6 hours and earns $48. Thus, the number of dollars Carola makes in 1 hour will be:

48/6 = $8

Therefore, the point which represents the number of dollars Carola makes per hour is (1, 8)

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Complete Question

Check attached image

The dimensions of the rectangle are 14 m by 7 m, and the minimum perimeter is: P = 14 + 2(7) = 28 m.

To find the minimum perimeter and dimensions of the enclosure, we can use the formula for the area of a rectangle: A = lw, where A is the area, l is the length, and w is the width.

Let's call the length of the rectangle x, and the width y. Since the fence will only have three sides, we know that one side of the rectangle will be the river. Therefore, the perimeter of the fence will be:

P = x + 2y

We want to minimize the perimeter, so we need to minimize the expression x + 2y. To do this, we can use the fact that the area is given as 98 m^2. We can set up an equation for the area in terms of x and y:

xy = 98

Now we can solve for one of the variables in terms of the other. Let's solve for y:

y = 98/x

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

P = x + 2(98/x)

To minimize P, we can take the derivative with respect to x and set it equal to zero:

dP/dx = 1 - 196/x^2 = 0

Solving for x, we get:

x = 14

Substituting this value for x into the equation for y, we get:

y = 7

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The probability of randomly selecting 3 plain M&M's in a row and having them all be brown is approximately 0.529%.

To calculate the probability of selecting 3 brown plain M&M's in a row, we need to first find the probability of selecting a brown M&M on the first pick, which is 13/52 (since there are 13 brown M&M's out of 52 total M&M's).

Then, for the second pick, there will be one less M&M in the bag and one less brown M&M, so the probability of selecting a brown M&M on the second pick is 12/51. Finally, for the third pick, there will be two less M&M's in the bag and two less brown M&M's, so the probability of selecting a brown M&M on the third pick is 11/50. To find the probability of all three events happening, we multiply the probabilities together:
(13/52) x (12/51) x (11/50) = 0.00529 or 0.529%

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1) The height of the arrow after two seconds is 27.8 meters

2) The time it takes for the arrow to hit the ground after it is fired is approximately 3.27 seconds.

3) The time when the arrow is 10 meters in the air is approximately 0.63 seconds.

1) To find the height of the arrow after 2 seconds, we need to substitute t = 2 in the equation:

h(2) = -9.8(2)² + 32(2) + 3

h(2) = -39.2 + 64 + 3

h(2) = 27.8

2) To find the time it takes for the arrow to hit the ground, we need to find the value of t when h(t) = 0. This is because when the arrow hits the ground, its height is zero. So we can set h(t) = 0 and solve for t:

-9.8t² + 32t + 3 = 0

Using the quadratic formula, we get:

t = (-32 + √(32² - 4(-9.8)(3))) ÷ (2(-9.8))

t = (-32 +√(1280.4)) ÷ (-19.6)

t = 3.27

3) To find the time when the arrow is 10 meters in the air, we need to solve the equation h(t) = 10 for t:

-9.8t² + 32t + 3 = 10

-9.8t² + 32t - 7 = 0

Using the quadratic formula, we get:

t = (-32 +√(32² - 4(-9.8)(-7))) ÷ (2(-9.8))

t = (-32 + √(1033.6)) ÷ (-19.6)

t = 0.63

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The complete question is:

An archer shoots an arrow up into the air the height h(t) in meters of the arrow after t seconds is modeled by h(t) = -9.8t^2 + 32t + 3

1) What is the height of the air after two seconds?

2) How long will it take for the air to hit the ground after it is fired?

3) At what time will the arrow be 10 m in the air?

To rewrite the linear system as a matrix equation, we can let y = [y1, y2, y3] and A be the coefficient matrix:

y' = Ay
where
A = [1 1 2; 1 0 1; 2 1 3]

To compute the eigenvalues of A, we can use the formula:
det(A - λI) = 0
where det represents the determinant and I is the identity matrix.

So, we have:
|1-λ 1 2|     |1 1-λ 2|     |1 1 1-λ|
|1 0-λ 1|  =  |1 0 1|  =  |1-λ 0 1|
|2 1 3-λ|     |2 1 3|     |2 1 3-λ|

Expanding the determinants, we get:
(1-λ)[(0-λ)(3-λ)-1]-1[(1)(3-λ)-2(1)]+2[(1)(1)-2(1-λ)]
= (λ-3)(λ-1)(λ-2) = 0

Therefore, the eigenvalues of A are 3, 1, and 2.

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Both B and C could be the type of function that exhibits the behavior, the average rate of change for a function is greater from x = 1 to x = 2 than

from x = 2 to x = 3.

Given that,

the average rate of change is greater from x = 1 to x = 2 than from x = 2 to x = 3, this implies that the slope of the function is steeper in the interval [1,2] than in the interval [2,3]. This means that the function is getting steeper as x increases, but the

rate of increase is decreasing.

The following types of functions can exhibit this

behavior:

A. Linear: A linear function has a constant slope, so it cannot exhibit this behavior.

B. Quadratic: A quadratic function has a changing slope, so it can exhibit this behavior. However, it depends on the specific coefficients of the quadratic equation.

C. Exponential: An exponential function can also exhibit this behavior. For example, y =

[tex]2^x[/tex]

has a steeper slope for smaller values of x, but the rate of increase slows down as x gets larger.

Therefore, both B and C could be the type of function that exhibits this behavior.

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Answer:

4

Step-by-step explanation:

If BD bisects <ABC, then that means it also bisects AC.

If a line is bisecting another line, then both parts are equal.  So, x has to be equal to 4.

Hope this helps :)

Answer: y-axis, because the x-coordinate is the opposite

Step-by-step explanation:

If we are reflecting a coordinate, the resulting coordinate(s) that end up as the opposite form of themselves correspond to the opposite axis. This means that if the x-coordinate is becoming opposite, we are reflecting over the y axis.

The energy required to change the separation between the centers of the basketballs to 13 m: 1.44 x 10^-12 J

To calculate the energy required to change the separation between the centers of the basketballs, we can use the formula for the potential energy of two point masses:

U = -G(m1m2)/r

where U is the potential energy, G is the gravitational constant, m1 and m2 are the masses of the basketballs, and r is the separation between their centers.

For the first case, where the separation between the centers is changed from the sum of their radii (0.28 m) to 1.1 m, we have:

r = 1.1 - 0.28 = 0.82 m

Plugging in the values, we get:

U = -6.67 x 10^-11 x 0.55^2 / 0.82 = -2.62 x 10^-10 J

Therefore, 2.62 x 10^-10 J of energy is required to change the separation between the centers of the basketballs to 1.1 m.

For the second case, where the separation between the centers is changed to 13 m, we have:

r = 13 - 0.28 = 12.72 m

Plugging in the values, we get:

U = -6.67 x 10^-11 x 0.55^2 / 12.72 = -1.44 x 10^-12 J

Therefore, 1.44 x 10^-12 J of energy is required to change the separation between the centers of the basketballs to 13 m.

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The change in angular momentum of the bolt is 11.25 kg m²/s.

Angular momentum is a measure of an object's tendency to keep rotating. It is calculated as the product of the object's moment of inertia and its angular velocity. The change in angular momentum is given by the torque applied to the object over a certain time interval. In this case, 45 Nm of torque is applied to a large bolt in 0.25s, and we need to find the change in angular momentum of the bolt.

To find the change in angular momentum, we need to first calculate the moment of inertia of the bolt. This depends on the shape and size of the bolt. Once we have the moment of inertia, we can use the formula:

Change in angular momentum = torque x time interval

Substituting the given values, we get:

Change in angular momentum = 45 Nm x 0.25 s = 11.25 Nms

Therefore, the change in angular momentum of the bolt is 11.25 Nms. This means that the bolt's tendency to rotate has increased by this amount due to the torque applied to it.

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