1. (100 pts) Consider two players A and B in a contest. The first one who wins a game wins the
contest. In the case of 5 successive draws, the contest is declared drawn. The probability that A
wins a game is 0.2. The probability that B wins a game is 0.5. And, thus, the probability of a draw
is 0.3. The winning or draw of each game is independent of previous games. The duration of the
contest is defined as the total number of games that are played before a winner is determined or
the contest is declared drawn.
(a) What is the PMF of the duration of the contest?
(b) What is the probability that the duration is at least 4?

The exact value of the trigonometric expression, given the conditions of sin and sec is -85/36.

The exact value of the trigonometric expression with u and v in Quadrant III is 304/425.

We are given sin(u) = -3/5 with 3π/2 < u < 2π and cos(v) = 15/17 with 0 < v < π/2.

cos(v - u) = cos(v)cos(u) + sin(v)sin(u)

We are given cos(v) = 15/17 and sin(u) = -3/5. To find sin(v) and cos(u), we can use the Pythagorean identities:

sin^2(v) + cos^2(v) = 1

sin(v) = sqrt(1 - (15/17)^2) = 8/17

Similarly, for cos(u):

sin^2(u) + cos^2(u) = 1

cos(u) = -sqrt(1 - (-3/5)^2) = -4/5

Now we can find cos(v - u):

cos(v - u) = cos(v)cos(u) + sin(v)sin(u)

cos(v - u) = -36/85

Since sec(v - u) = 1/cos(v - u), we have:

sec(v - u) = 1/(-36/85) = -85/36

Since both u and v are in Quadrant III, sin(u) and cos(u) are both negative, and sin(v) and cos(v) are both negative. We are given sin(u) = -7/25 and cos(v) = -15/17.

To find sin(v) and cos(u), we can use the Pythagorean identities:

sin^2(u) + cos^2(u) = 1

cos(u) = -sqrt(1 - (-7/25)^2) = -24/25 (since u is in Quadrant III)

Similarly, for sin(v):

sin^2(v) + cos^2(v) = 1

sin(v) = -sqrt(1 - (-15/17)^2) = -8/17 (since v is in Quadrant III)

Now we can find cos(u + v):

cos(u + v) = (-24/25)(-15/17) - (-7/25)(-8/17)

cos(u + v) = 304/425

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The interest is $9.To find the interest, we can use the simple interest formula:

I = P * r * t

what is interest ?

Interest is the amount of money charged by a lender to a borrower for the use of money, usually expressed as a percentage of the amount borrowed. In other words, it is the cost of borrowing money.

In the given question,

To find the interest, we can use the simple interest formula:

I = P * r * t

where I is the interest, P is the principal, r is the annual interest rate as a decimal, and t is the time in years.

Here, the principal is $200, the annual interest rate is 9%, and the time is 1/2 year.

Converting the annual interest rate to a decimal, we get:

r = 9% = 0.09

And, converting the time to years, we get:

t = 1/2 year

Substituting the values into the formula, we get:

I = $200 * 0.09 * (1/2) = $9

Therefore, the interest is $9.

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Answer:1152

Step-by-step explanation:monkey

It would cost $282.60 to fill a cylindrical tank with a diameter of 6 feet and a height of 2 feet.

A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases of equal size and shape, connected by a curved side.

To calculate the cost of filling a tank with a diameter of 6 feet and a height of 2 feet, assuming a cost of $5 per cubic foot and using the value of pi as 3.14, we first need to calculate the volume of the tank.

The volume of a cylinder (which is the shape of the tank) is given by the formula:

V = π[tex]r^2h[/tex]

where V is the volume, r is the radius (half the diameter), and h is the height.

Since the diameter of the tank is 6 feet, the radius is 3 feet. Therefore:

V = 3.14 x [tex]3^2[/tex] x 2

= 56.52 cubic feet

Multiplying the volume of the tank by the cost per cubic foot gives us the total cost of filling the tank:

Total cost = 56.52 x $5

= $282.60

Therefore, it would cost $282.60 to fill a tank with a diameter of 6 feet and a height of 2 feet, assuming a cost of $5 per cubic foot and using the value of pi as 3.14.

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Answer:

2,000

Step-by-step explanation:

1/6

Step-by-step explanation:

the reason is because of the tail and the spinner

The eigen values of the first matrix is λ=4, λ=0 and λ=-3. The eigenvalues of the second matrix is 0 and 6.

If there is a scalar such that Ax = x, then a non-zero vector x is an eigenvector of A given a square matrix A. The scalar eigenvalue of the eigenvector x is known as the scalar.

To put it another way, when an eigenvector x is added to a matrix A, the outcome is a scalar multiple of x, where the scalar is the eigenvalue.

Many applications of linear algebra depend on eigenvectors and eigenvalues, such as the solution of differential equation systems, identifying the major variables in a dataset, and assessing the stability of dynamic systems.

The eigenvalues of the matrix can be found using the equation:

det(A - λI) = 0

Substituting the values we have:

[tex]\left[\begin{array}{ccc}4&2&0\\0&0&0\\1&0&-3\end{array}\right] - \lambda \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc} 4 - \lambda &0&0\\0&- \lambda & 0\\1 &0 & -3 - \lambda\end{array}\right][/tex]

Now determine the det of the matrix:

(4 - λ) (-3λ + λ² ) - 0 + 0 = (4 - λ)(λ)(-3 + λ)

Thus the eigen values are:

λ= 4, λ = 0, and λ = 3.

For the second matrix we have:

Given that B has identical rows throughout, B has a rank of 1, and B's null space has a dimension of 2. With two eigenvalues equal to zero and one eigenvalue equal to the trace of B, which is 6, B has two eigenvalues equal to zero.

Hence, the eigenvalues of B are 0 and 6.

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Answer:

yes

Step-by-step explanation:

Answer:

Step-by-step explanation:

Here, a a⁻¹ b = 1b = b shows that a⁻¹ is indeed inverse of a, and x = a⁻¹b is a valid solution to the equation ax = b.

An equation is a statement that two expressions are equal. A solution of an equation is a value (or a set of values) that makes the equation true.

In the equation ax = b, the solution x = a⁻¹b means that if we multiply a by its inverse a⁻¹, we get 1, the multiplicative identity. So, when we multiply both sides of the equation by a⁻¹, we get:

a⁻¹(ax) = a⁻¹b

Multiplying a⁻¹ and a on the left side of the equation gives:

1x = a⁻¹b

which simplifies to x = a⁻¹b. This shows that x = a⁻¹b is a solution to the equation ax = b.

The statement "a a⁻¹ b = 1b = b" means that when we multiply a by its inverse a⁻¹, we get the multiplicative identity 1, and when we multiply 1 by b, we get b. So, a a⁻¹ b = 1b = b shows that a⁻¹ is indeed the inverse of a, and x = a⁻¹b is a valid solution to the equation ax = b.

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Justin is 14 years old, and Hassan is 15 years old.

What are integers?

The Latin term "Integer," which implies entire or intact, is where the word "integer" first appeared. Zero, positive numbers, and negative numbers make up the particular set of numbers known as integers.

Let's assume that Justin's age is x. Then, Hassan's age is x + 1, since their ages are consecutive integers and Hassan is older than Justin.

We are given that the product of their ages is 210:

x(x + 1) = 210

Expanding the left-hand side and simplifying, we get:

x² + x - 210 = 0

This is a quadratic equation in standard form. We can solve for x by using the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = 1, and c = -210.

Plugging these values into the formula, we get:

x = (-1 ± √(1² - 4(1)(-210))) / 2(1)

x = (-1 ± √(1 + 840)) / 2

x = (-1 ± √(841)) / 2

We take the positive root because x represents Justin's age, which is a positive integer. Therefore:

x = (-1 + 29) / 2 = 14

So, Justin is 14 years old, and Hassan is 15 years old.

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Answer: 75%

Step-by-step explanation:

Therefore, the overall difference in loss or gain between selling at the current day's high price or low price would be $3460.

An expression is a mathematical or logical statement that represents a value or a result. It can consist of numbers, variables, operators, and/or functions that are combined in a specific way to form a valid statement. Expressions can be simple, such as a single number or variable, or complex, involving multiple operations and functions. In mathematics, expressions can be used to represent mathematical operations such as addition, subtraction, multiplication, division, exponentiation, and more.

Here,

To calculate the difference in overall loss or gain between selling at the current day's high price or low price for the stocks purchased, we need to know the current day's high and low prices for stock A and stock B. Once we have that information, we can use the following formula:

Difference = (Current day's high price - Purchase price) * Number of shares

Let's assume the current day's high price for stock A is $100 per share and the low price is $80 per share, while the high price for stock B is $150 per share and the low price is $130 per share.

For stock A:

Purchase price = $90 per share

Number of shares = 120 shares

Difference in overall loss or gain for stock A when selling at the high price:

Difference = ($100 - $90) * 120

= $1200

Difference in overall loss or gain for stock A when selling at the low price:

Difference = ($80 - $90) * 120

= -$1080

For stock B:

Purchase price = $145 per share

Number of shares = 35 shares

Difference in overall loss or gain for stock B when selling at the high price:

Difference = ($150 - $145) * 35

= $175

Difference in overall loss or gain for stock B when selling at the low price:

Difference = ($130 - $145) * 35

= -$525

So, the overall difference in loss or gain between selling at the current day's high price or low price would be the sum of the differences for stock A and stock B:

Overall difference in loss or gain = (Difference for stock A when selling at the high price + Difference for stock B when selling at the high price) - (Difference for stock A when selling at the low price + Difference for stock B when selling at the low price)

Overall difference in loss or gain = ($1200 + $175) - (-$1080 + -$525)

= $1855 + $1605

= $3460

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Table A is showing the correct relation of equation y=1.5x

An equation is a mathematical statement that asserts that two expressions are equal. It typically consists of two parts: the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=). They are used to model real-world phenomena, to solve problems, and to make predictions. Examples of equations include linear equations, quadratic equations, systems of equations, and differential equations.

Using the relation;

y=1.5x

Putting the value x=9

y=1.5×9

y=13.5

Putting the value x=11

y=1.5×11

y=16.5

On observing the tables, option A is satisfying the relation

Hence, Table A is showing the correct relation of equation y=1.5x.

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The complete question is:

Image is attached below.

The equation of the graph is

The table is completed by substituting the x to the function y = sqrt(x) + 2

When x = 0:

y = sqrt(0) + 2

y = 0 + 2

y = 2

When x = 1:

y = sqrt(1) + 2

y = 1 + 2

y = 3

When x = 4:

y = sqrt(4) + 2

y = 2 + 2

y = 4

When x = 9:

y = sqrt(9) + 2

y = 3 + 2

y = 5

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The mean of the set is 1/2.

The mean is a statistical measure that represents the average value of a set of numbers. To calculate the mean, you add up all the values in the set and then divide by the total number of values in the set.

To find the mean of a set of numbers, you add up all the numbers in the set and then divide the result by the total number of numbers in the set.

In this case, the set has six numbers:

2/3, 1/6, 1/3, 2/3, 7/9, 1/3

Adding them up, we get:

(2/3) + (1/6) + (1/3) + (2/3) + (7/9) + (1/3) = 3

So the sum of the numbers in the set is 3.

To find the mean, we divide this sum by the total number of numbers in the set, which is 6:

3/6 = 1/2

Therefore, the mean of the set is 1/2.

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Answer: 1,024 different ways

Step-by-step explanation:

      We will use the Fundamental Counting Principle. There are 5 questions, each with 4 possible answers. To use the principle and answer the question, we will multiply 4 by itself 5 times.

             4 * 4 * 4 * 4 * 4 = 1,024 different ways

Think of it like this:

            you have 4 objects (the 4 answer options, a, b, c, and d)

            You are going to chose a group of 5, and the order matters (your         answer to question #1 is independent of your choice on #3)

             Repetition is allowed (you can answer "c" on all of the questions if you want to)

This is a simple permutation. Use the formula:

       [tex]\bold{n^r}[/tex]

       where n is the number of objects (the number of answer choices)

       and r is the number of objects you will choose (5, one answer for each question)

To solve:

[tex]4^5[/tex]

[tex]1,024 \longleftarrow[/tex] your answer!

Hope this helps! Feel free to ask any follow up questions, and please "vote up" :)

The possible solutions to the equation are:

3π/4

π/4

Trigonometric equations involve trigonometric functions such as sine, cosine, tangent, etc. The goal is to solve for the unknown variable in the equation.

csc²x - 2 = 0

csc²x = 2

csc x = √2

sin x = 1/√2    (Remember: csc x  = 1/sinx)

x = sin⁻¹(1/√2 )

x = 90° or 135°  (sine is positive in 1st and 2nd quadrants)

x = π/4 or 3π/4

Thus, the possible solutions to the equation are x = π/4 or 3π/4.

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The potential rational zeros of the given polynomial are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±15, ±20, ±30, ±60, ±1/2, ±1/3, ±2/3, ±5/2, ±4/3, ±5/3, ±10/3, ±1/6, ±5/6, ±1/10, ±2/5, ±4/15, ±1/15, ±2/15.

How to calculate potential rational zeroes?

The Rational Zero Theorem states that if a polynomial with integer coefficients has any rational zeros, then they must have the form of a fraction p/q, where p is a constant term factor and q is a leading coefficient factor.

For the given polynomial q(x)=6x^3+31x^2+23x−20, the constant term is -20, and the leading coefficient is 6. Therefore, the potential rational zeros can be expressed as follows:

p/q = ± {factors of the constant term (-20)}/{factors of the leading coefficient (6)}

Possible factors of -20: ±1, ±2, ±4, ±5, ±10, ±20

Possible factors of 6: ±1, ±2, ±3, ±6

Therefore, the potential rational zeros are:

±1/1, ±2/1, ±4/1, ±5/1, ±10/1, ±20/1,

±1/2, ±2/2, ±4/2, ±5/2, ±10/2, ±20/2,

±1/3, ±2/3, ±4/3, ±5/3, ±10/3, ±20/3,

±1/6, ±2/6, ±4/6, ±5/6, ±10/6, ±20/6.

Simplifying and eliminating duplicates, we get:

±1, ±2, ±4, ±5, ±10, ±20,

±1/2, ±2/2, ±4/2, ±5/2, ±10/2, ±20/2 (which simplifies to ±1, ±2, ±3, ±5, ±10, ±20),

±1/3, ±2/3, ±4/3, ±5/3, ±10/3, ±20/3 (which simplifies to ±1/3, ±2/3, ±4/3, ±5/3, ±10/3),

±1/6, ±2/6, ±4/6, ±5/6, ±10/6, ±20/6 (which simplifies to ±1/6, ±1/3, ±2/3, ±5/6, ±5/3, ±10/3).

Therefore, the potential rational zeros of the given polynomial are ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±15, ±20, ±30, ±60, ±1/2, ±1/3, ±2/3, ±5/2, ±4/3, ±5/3, ±10/3, ±1/6, ±5/6, ±1/10, ±2/5, ±4/15, ±1/15, ±2/15.

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After three years, Carlo is taller than Reggie. Carlo is 179 cm tall and Reggie is 175 cm tall.

A method for measuring temperature, length, volume, weight, and distance is known as the metric system. It is founded on three fundamental units that allow us to measure almost anything in the universe.

To compare the heights of Carlo and Reggie, we need to convert their heights to the same unit. We can convert Reggie's height from millimeters to centimeters by dividing by 10:

Reggie's height = 1680 mm ÷ 10 = 168 cm

Now we can compare their heights before and after three years:

After three years:

Therefore, after three years, Carlo is taller than Reggie. Carlo is 179 cm tall and Reggie is 175 cm tall.

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Answer:

Step-by-step explanation:

Applying absolute value rule the solutions are:

x + 1/2 = 3/2/3       or        x + 1/2 = -3 2/3

Changing to fractions:

x + 1/2 = 11/3   or   x + 1/2 = -11/3

and then changing to a common denominator

x + 3/6 = 22/6    or    x + 3/6  = -22/6

Solve left equation first:

x + 3/6 - 3/6 = 22/6 - 3/6

x = 19/6 or 3 1/6

Solve the right equation:

x + 3/6 - 3/6 = -22/6 - 3/6

x = -25/6 or  -4 1/6

So the two solutions are:

19/6 and -25/6 as improper fractions

And

as mixed numbers:  3 1/6 and -4 1/6

The value of X in the given fraction is 1/12

A fraction is a mathematical expression that represents a part of a whole or a quotient of two numbers. It is expressed as one integer or number (called the numerator) divided by another integer or number (called the denominator), separated by a line or a slash.

Here, we are going to find the LCM of both 4 and 6 to determine the value of X.

LCM = 12

X = 1/4 - 1/6

X = (3 - 2)/12

X = 1/12

Therefore, we can conclude that the value of X in the given fraction is 1/12.

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Question 1:

(a) To find the probability that a taxi driver earns less than $1500 in a day, we need to standardize the value using the given mean and standard deviation, and then find the corresponding probability from the standard normal distribution table:

z = (1500 - 1062.5) / 350 = 1.20

Using the standard normal distribution table, the probability of a standard normal random variable being less than 1.20 is approximately 0.8849. Therefore, the probability that a taxi driver earns less than $1500 in a day is approximately:

P(X < 1500) = P(Z < 1.20) = 0.8849

(b) We need to find the value of k such that 93.7% of the drivers earn more than $k a day. This means that the probability of a driver earning less than or equal to $k a day is 1 - 0.937 = 0.063. We can standardize k using the given mean and standard deviation, and then find the corresponding z-score from the standard normal distribution table:

z = (k - 1062.5) / 350

Using the standard normal distribution table, we find that the z-score corresponding to a probability of 0.063 is approximately -1.51. Therefore:

-1.51 = (k - 1062.5) / 350

k = -1.51 * 350 + 1062.5 = $499.25 (rounded to the nearest cent)

(c) The mean of the total daily earning is:

μT = μ1 + μ2 = 1062.5 + 235.5 = 1298

The variance of the total daily earning is the sum of the variances of the two earnings, since they are assumed to be independent:

σT² = σ1² + σ2² = 350² + 84.5² ≈ 128681

Therefore, the standard deviation of the total daily earning is:

σT ≈ √128681 ≈ 358.5

(rounded to the nearest integer)

(d) To find L1 and L2, we need to find the z-scores corresponding to the lower and upper 2.2% tails of the standard normal distribution:

z1 = -1.81

z2 = 1.81

Then we can use the formula for standardizing a normal random variable to find the corresponding values of T:

z1 = (L1 - μT) / σT

z2 = (L2 - μT) / σT

Solving for L1 and L2, we get:

L1 = μT + z1σT ≈ 1298 + (-1.81) * 358.5 ≈ $645

L2 = μT + z2σT ≈ 1298 + 1.81 * 358.5 ≈ $1951

(rounded to the nearest integer)

Question 2:

(a) We can model the transaction time of a single customer as a normal random variable with mean 20 and standard deviation 5. Then the total transaction time for 6 customers can be modeled as a normal random variable with mean 6 * 20 = 120 and standard deviation √(6 * 5²) = 15. To find the probability that 5 customers can finish the transaction within 20 seconds, we need to standardize the value using this mean and standard deviation, and then find the corresponding probability from the standard normal distribution table:

z = (5 * 20 - 120) / 15 = -0.53

The fifth multiple of each fraction is shown as follows:

1. 2/6 = 5/3

2. 7/10 = 7/2

3. 3/8 = 15/8

4. 5/12 =25/12

5. 7/6 = 35/6

To get the multiple of a fraction, the best thing to do will be to multiply the fraction by the whole number in question. In this case, the whole number is five.

So, to get the right answers, you can begin by multiplying the functions with the whole number five. The values obtained are the fifth multiples. for the fraction, 2/6, the fifth multiple is 5/3.

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Answer: Factors of a polynomial is x(x+1)

Step-by-step explanation: To find the GCF, we first seek the greatest common divisor of these two terms.

Taking common x from (x^2+x).

x(x+1)

The total number of college students is 204,700,000.

Percentage, a relative value indicating hundredth parts of any quantity. One percent (symbolized 1%) is a hundredth part; thus, 100 percent represents the entirety and 200 percent specifies twice the given quantity.

Here, Percentage of enrolled student for part time = 23%

         Total enrolled students = 8.9 millions

Number of part time student = 23% x 8.9 million

                                              [tex]= 0.23 \times 8900000[/tex]

                                             [tex]=204,700,000[/tex]

Thus, the total number of college students is 204,700,000.

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Overall, the Perez family likely benefited both financially and emotionally from each of these sources during their time of need.

The Perez family likely benefited in different ways from each of these sources during their time of need:

Insurance: If the Perez family had insurance, it is likely that they were able to receive financial assistance to cover some of their expenses. Depending on the type of insurance they had, they may have been able to receive money to pay for medical bills, property damage, or other expenses related to their time of need.

Non-profits: Non-profits may have provided the Perez family with resources such as food, clothing, and shelter during their time of need. These organizations may have also provided emotional support to the family, helping them feel less alone during a difficult time.

Family and friends: Family and friends likely provided the Perez family with emotional support during their time of need, as well as practical assistance such as meals, childcare, and help with household tasks. Financially, family and friends may have also helped the Perez family by providing loans or gifts of money.

Government: The government may have provided the Perez family with financial assistance through programs such as unemployment benefits, food stamps, or housing assistance. Additionally, the government may have provided emotional support to the family through counseling services or other resources.

Overall, the Perez family likely benefited both financially and emotionally from each of these sources during their time of need.

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Answer:

Sequence A is defined by subtracting 6 from the previous term and taking the absolute value, while Sequence B is the sum of the first n odd integers. B(9) > A(9).

To produce each new term from the previous term, subtract 6 from the previous term, then take the absolute value of the result. That is, if the previous term is x, then the next term is |x - 6|.

To find A(9) and B(9), we can apply the rules we have for each sequence:

A(0) = 19

A(1) = |19 - 6| = 13

A(2) = |13 - 6| = 7

A(3) = |7 - 6| = 1

A(4) = |1 - 6| = 5

A(5) = |5 - 6| = 1

A(6) = |1 - 6| = 5

A(7) = |5 - 6| = 1

A(8) = |1 - 6| = 5

A(9) = |5 - 6| = 1

B(0) = 0 (the sum of the first 0 odd integers)

B(1) = 1

B(2) = 1 + 3 = 4

B(3) = 1 + 3 + 5 = 9

B(4) = 1 + 3 + 5 + 7 = 16

B(5) = 1 + 3 + 5 + 7 + 9 = 25

B(6) = 1 + 3 + 5 + 7 + 9 + 11 = 36

B(7) = 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49

B(8) = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

B(9) = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = 81

Answer:

Step-by-step explanation:

d.