The Statistical Abstract of the United States published by the U.S. Census Bureau reports that the average annual consumption of fresh fruit per person is 99.9 pounds. The standard deviation of fresh fruit consumption is about 30 pounds. Suppose a researcher took a random sample of 38 people and had them keep a record of the fresh fruit they ate for one year.
Appendix A Statistical Tables
(Round all z values to 2 decimal places. Round your answers to 4 decimal places.)
a. What is the probability that the sample average would be less than 90 pounds?
p =
b. What is the probability that the sample average would be between 98 and 105 pounds?
p =
c. What is the probability that the sample average would be less than 112 pounds?
p =
d. What is the probability that the sample average would be between 93 and 96 pounds?
p =

Answer: P = 4s

Step-by-step explanation:

P = 4s where s = the length of each side.  

Since each side of a square is the same length, the side length is multiplied by 4.

Given:

The average number of songs performed there in a 10 day period is 167.

To find:

The number of songs performed there in a year time.

Solution:

We have,

Number of songs performed in 10 days = 167

Number of songs performed in 1 day = [tex]\dfrac{167}{10}[/tex]

                                                              = [tex]1.67[/tex]

We know that 1 year is equal to 365 days. So,

Number of songs performed in 365 day = [tex]1.67\times 365[/tex]

Number of songs performed in 1 year     = [tex]609.55[/tex]

                                                                   [tex]\approx 610[/tex]

Therefore, the number of songs performed there in a year time is about 610.

Answer:

0.7588 = 75.88% probability that more than 1 vessel transporting nuclear weapons was destroyed

Step-by-step explanation:

The vessels are destroyed without replacement, which means that the hypergeometric distribution is used to solve this question.

Hypergeometric distribution:

The probability of x successes is given by the following formula:

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

In which:

x is the number of successes.

N is the size of the population.

n is the size of the sample.

k is the total number of desired outcomes.

Combinations formula:

[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

In this question:

Fleet of 17 means that [tex]N = 17[/tex]

4 are carrying nucleas weapons, which means that [tex]k = 4[/tex]

9 are destroyed, which means that [tex]n = 9[/tex]

What is the probability that more than 1 vessel transporting nuclear weapons was destroyed?

This is:

[tex]P(X > 1) = 1 - P(X \leq 1)[/tex]

In which

[tex]P(X \leq 1) = P(X = 0) + P(X = 1)[/tex]

So

[tex]P(X = x) = h(x,N,n,k) = \frac{C_{k,x}*C_{N-k,n-x}}{C_{N,n}}[/tex]

[tex]P(X = 0) = h(0,17,9,4) = \frac{C_{4,0}*C_{13,9}}{C_{17,9}} = 0.0294[/tex]

[tex]P(X = 1) = h(1,17,9,4) = \frac{C_{4,1}*C_{13,8}}{C_{17,9}} = 0.2118[/tex]

Then

[tex]P(X \leq 1) = P(X = 0) + P(X = 1) = 0.0294 + 0.2118 = 0.2412[/tex]

[tex]P(X > 1) = 1 - P(X \leq 1) = 1 - 0.2412 = 0.7588[/tex]

0.7588 = 75.88% probability that more than 1 vessel transporting nuclear weapons was destroyed

Answer:

The sum of squared errors for the regression equation is 62.

Step-by-step explanation:

The sum of squared errors can be computed as follows:

X            Y           Y* = 2 + 3X         Y - Y*           (Y - Y*)^2

0            5                  2                      3                    9

3            5                  11                     -6                  36

7            27               23                     4                    16

10          31                32                    -1                     1  

20         68               68                     0                   62

From the above, we have:

Error = Y -  Y*

Error^2 = (Y - Y*)^2

Sum of squared errors = Sum of Error^2 = Total of (Y - Y*)^2 = 62

Therefore, the sum of squared errors for the regression equation is 62.

Step-by-step explanation:

always Pythagoras with the coordinate differences as sides and the distance the Hypotenuse.

c² = (2/3 - 5/3)² + (-10/9 - -7/9)² = (-3/3)² + (-10/9 + 7/9)² =

= (-1)² + (-3/9)² = 1 + (-1/3)² = 1 + 1/9 = 10/9

c = sqrt(10)/3

Answer:

Step-by-step explanation:

Point 1  ([tex]\frac{2}{3}[/tex] , [tex]\frac{-10}{9}[/tex])   in the form (x1,y1)

Point 2 ( [tex]\frac{5}{3}[/tex] , [tex]\frac{-7}{9}[/tex])  in the form (x2,y2)

use the distance formula

dist = sqrt[ (x2-x1)^2 + (y2-y1)^2 ]

dist = sqrt [ [tex]\frac{5}{3}[/tex] -[tex]\frac{2}{3}[/tex])^2 + (  [tex]\frac{-7}{9}[/tex] - ( [tex]\frac{-10}{9}[/tex] ) )^2 ]

dist = sqrt [ ([tex]\frac{3}{3}[/tex])^2 + ([tex]\frac{3}{9}[/tex])^2 ]

dist = sqrt [  1 + ([tex]\frac{1}{3}[/tex])^2 ]

dist = sqrt [  [tex]\frac{9}{9}[/tex] + [tex]\frac{1}{9}[/tex] ]

dist = [tex]\sqrt{\frac{10}{9} }[/tex]

dist = [tex]\sqrt{10}[/tex] *[tex]\sqrt{\frac{1}{9} }[/tex]

dist = [tex]\sqrt{10}[/tex]  * [tex]\frac{1}{3}[/tex]

dist = [tex]\frac{\sqrt{10} }{3}[/tex]

Answer:

180

Step-by-step explanation:

Given the cost (in dollars) of buying x pounds of a party product is given by the function

C(x) = 10x + 300.

Suppose, for budgetary reasons; you can't spend more than $2100 on this product. You can spend less, but you have to buy at least 50 pounds.

To get the domain of the function, substitute C(x) =2100 and find x

2100 = 10x + 300

10x = 2100 - 300

10x = 1800

x = 1800/10

x = 180

Hence the domain of the function is 180

Answer:

157.8

Step-by-step explanation:

Add them all up to get 789 and divide them by 5 as there are five numbers to get the answer:)

Answer:

0.0207 = 2.07% approximate probability of finding at least 157 defects

Step-by-step explanation:

Poisson distribution:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

[tex]P(X = x) = \frac{e^{-\lambda}*\lambda^{x}}{(x)!}[/tex]

In which

x is the number of sucesses

e = 2.71828 is the Euler number

[tex]\lambda[/tex] is the mean in the given interval.

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

n instances of a Poisson distribution can be approximated to a normal distribution, with [tex]\mu = n\lambda, \sigma = \sqrt{\lambda}\sqrt{n}[/tex]

The average defect rate on a 2010 Volkswagen vehicle was reported to be 1.33 defects per vehicle.

This means that [tex]\lambda = 1.33[/tex]

Suppose that we inspect 100 Volkswagen vehicles at random.

This means that [tex]n = 100[/tex]

Mean and standard deviation:

[tex]\mu = n\lambda = 100*1.33 = 133[/tex]

[tex]\sigma = \sqrt{\lambda}\sqrt{n} = \sqrt{1.33}\sqrt{100} = 11.53[/tex]

What is the approximate probability of finding at least 157 defects?

Using continuity correction(Poisson is a discrete distribution, normal continuous), this is [tex]P(X \geq 157 - 0.5) = P(X \geq 156.5)[/tex], which is 1 subtracted by the p-value of Z when X = 156.5. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{156.5 - 133}{11.53}[/tex]

[tex]Z = 2.04[/tex]

[tex]Z = 2.04[/tex] has a p-value of 0.9793.

1 - 0.9793 = 0.0207

0.0207 = 2.07% approximate probability of finding at least 157 defects

Answer:

C=25°

a=11

b=12

Step-by-step explanation:

Find angle c,since angles in a triangle add up to 180 and we know angleA andB angle C will be

65+90+C=180

C=180-155

C=25°

To find a

use trig ratios

tanA=opposite/adjacent

tan65=a/5

a=tan65×5

a=10.72 round off to 11

To find b

sinC=opposite/hypotenuse

sin25=5/b

sin25 b=5

b=11.8 or rather 12

Answer:

Step-by-step explanation:

First find  side a and to find this  calculate tan 65

Tan 65 = [tex]\frac{opposite \ side}{adjacent\ side}=\frac{a}{5}\\\\[/tex]

2.144 = a/5

a = 2.144 * 5

b² = a² + c²

   = 121+25

   = 146.

b = √146 = 12.08 = 12

a = 10.72 = 11

Now find Tan C

[tex]Tan \ C = \frac{5}{10.72}\\\\Tan \ C = 0.4664\\[/tex]

C = tan⁻¹ 0.4664

C = 25°

9514 1404 393

Answer:

  ≈ 1000√10∠-129.04464° = -1992 -2456i

Step-by-step explanation:

  3 -i = √(3³+(-1)²)∠arctan(-1/3) ≈ √10∠-18.4349°

Then (3-i)^7 = 10^(7/2)∠(7×-18.4349°) = 1000√10∠-129.04464°

  = 1000√10(cos(-129.04464°) +i·sin(-129.04464°))

  = -1992 -2456i

Answer:

a) 1186

b) Between 1031 and 1493.

c) 160

Step-by-step explanation:

Normal Probability Distribution

Problems of normal distributions can be solved using the z-score formula.

In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Normally distributed with mean of 1262 and a standard deviation of 118.

This means that [tex]\mu = 1262, \sigma = 118[/tex]

a) Determine the 26th percentile for the number of chocolate chips in a bag. ​

This is X when Z has a p-value of 0.26, so X when Z = -0.643.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-0.643 = \frac{X - 1262}{118}[/tex]

[tex]X - 1262 = -0.643*118[/tex]

[tex]X = 1186[/tex]

(b) Determine the number of chocolate chips in a bag that make up the middle 95% of bags.

Between the 50 - (95/2) = 2.5th percentile and the 50 + (95/2) = 97.5th percentile.

2.5th percentile:

X when Z has a p-value of 0.025, so X when Z = -1.96.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-1.96 = \frac{X - 1262}{118}[/tex]

[tex]X - 1262 = -1.96*118[/tex]

[tex]X = 1031[/tex]

97.5th percentile:

X when Z has a p-value of 0.975, so X when Z = 1.96.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]1.96 = \frac{X - 1262}{118}[/tex]

[tex]X - 1262 = 1.96*118[/tex]

[tex]X = 1493[/tex]

Between 1031 and 1493.

​(c) What is the interquartile range of the number of chocolate chips in a bag of chocolate chip​ cookies?

Difference between the 75th percentile and the 25th percentile.

25th percentile:

X when Z has a p-value of 0.25, so X when Z = -0.675.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]-0.675 = \frac{X - 1262}{118}[/tex]

[tex]X - 1262 = -0.675*118[/tex]

[tex]X = 1182[/tex]

75th percentile:

X when Z has a p-value of 0.75, so X when Z = 0.675.

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]0.675 = \frac{X - 1262}{118}[/tex]

[tex]X - 1262 = 0.675*118[/tex]

[tex]X = 1342[/tex]

IQR:

1342 - 1182 = 160

Answer:

No it is not

Step-by-step explanation:

The median is 84200

The mean is 85300

Low income is 65100

High income is 103400

From this information, we can see a skewed data. The mean would not be a good estimate value. Rather the center (median) would be more appropriate.

When we calculate the middle value for this data

65100+103400 = 168500/2 = 84250

84250 is closer to the median score of 84200. The median is best in the presence of outliers.

Answer:

0.5015 = 50.15% probability that it came from manufacturer A.

Step-by-step explanation:

Conditional Probability

We use the conditional probability formula to solve this question. It is

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]

In which

P(B|A) is the probability of event B happening, given that A happened.

[tex]P(A \cap B)[/tex] is the probability of both A and B happening.

P(A) is the probability of A happening.

In this question:

Event A: Defective

Event B: From manufacturer A.

Probability a unit is defective:

2% of 43%(from manufacturer A)

1.5% of 57%(from manufacturer B). So

[tex]P(A) = 0.02*0.43 + 0.015*0.57 = 0.01715[/tex]

Probability a unit is defective and from manufacturer A:

2% of 43%. So

[tex]P(A \cap B) = 0.02*0.43 = 0.0086[/tex]

What is the probability that it came from manufacturer A?

[tex]P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{0.0086}{0.01715} = 0.5015[/tex]

0.5015 = 50.15% probability that it came from manufacturer A.

Answer:

TS = 12.7

Step-by-step explanation:

From the question given above, the following data were obtained:

QR = 9

QP = 6

TU = 19

TS =?

Since the triangles are SIMILAR, then,

QR / TU = QP / TS

With the above equation, we can obtain the value of TS as follow:

QR = 9

QP = 6

TU = 19

TS =?

QR / TU = QP / TS

9 / 19 = 6 / TS

Cross multiply

9 × TS = 19 × 6

9 × TS = 114

Divide both side by 9

TS = 114 / 9

TS = 12.7

The options are;

1) AB is bisected by CD

2) CD is bisected by AB

3) AE = 1/2 AB

4) EF = 1/2 ED

5) FD= EB

6) CE + EF = FD

Answer:

Options 1, 3 & 6 are correct

Step-by-step explanation:

We are told that Point E is the midpoint of AB. Thus, any line that passes through point E will bisect AB into two equal parts.

The only line passing through point E is line CD.

Thus, we can say that line AB is bisected by pine CD. - - - (1)

Also, since E is midpoint of Line AB, it means that;

AE = EB

Thus, AE = EB = ½AB - - - (2)

Also, we are told that F is the mid-point of CD.

Thus;

CF = FD

Point E lies between C and F.

Thus;

CE + EF = CF

Since CF =FD

Thus;

CE + EF = FD - - - (3)

Some symbols and numbers are missing. I assume the system is supposed to read

2x - y + 2z + w = -3

x + 2y - 3z + w = 12

3x - y - z + 2w = 3

-2x + 3y + 2z - 3w = -3

In matrix form, this is

[tex]\begin{bmatrix}2&-1&2&1\\1&2&-3&1\\3&-1&-1&2\\-2&3&2&-3\end{bmatrix}\begin{bmatrix}x\\y\\z\\w\end{bmatrix}=\begin{bmatrix}-3\\12\\3\-3\end{bmatrix}[/tex]

which we can strip down to the augmented matrix,

[tex]\left[\begin{array}{cccc|c}2&-1&2&1&-3\\1&2&-3&1&12\\3&-1&-1&2&3\\-2&3&2&-3&-3\end{array}\right][/tex]

Now for the row operations:

• swap rows 1 and 2

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\2&-1&2&1&-3\\3&-1&-1&2&3\\-2&3&2&-3&-3\end{array}\right][/tex]

• add -2 (row 1) to row 2, -3 (row 1) to row 3, and 2 (row 1) to row 4

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&-7&8&-1&-33\\0&7&-4&-1&21\end{array}\right][/tex]

• add 7 (row 2) to -5 (row 3), and row 3 to row 4

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&16&-2&-24\\0&0&4&-2&-12\end{array}\right][/tex]

• multiply through rows 3 and 4 by 1/2

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&8&-1&-12\\0&0&2&-1&-6\end{array}\right][/tex]

• add -4 (row 4) to row 3

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&0&3&12\\0&0&2&-1&-6\end{array}\right][/tex]

• swap rows 3 and 4

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&2&-1&-6\\0&0&0&3&12\end{array}\right][/tex]

• multiply through row 4 by 1/3

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&2&-1&-6\\0&0&0&1&4\end{array}\right][/tex]

• add row 4 to row 3

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&2&0&-2\\0&0&0&1&4\end{array}\right][/tex]

• multiply through row 3 by 1/2

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&8&-1&-27\\0&0&1&0&-1\\0&0&0&1&4\end{array}\right][/tex]

• add -8 (row 3) and row 4 to row 2

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&-5&0&0&-15\\0&0&1&0&-1\\0&0&0&1&4\end{array}\right][/tex]

• multiply through row 2 by -1/5

[tex]\left[\begin{array}{cccc|c}1&2&-3&1&12\\0&1&0&0&3\\0&0&1&0&-1\\0&0&0&1&4\end{array}\right][/tex]

• add -2 (row 2) and 3 (row 3) and -1 (row 4) to row 1

[tex]\left[\begin{array}{cccc|c}1&0&0&0&-1\\0&1&0&0&3\\0&0&1&0&-1\\0&0&0&1&4\end{array}\right][/tex]

Then the solution to the system is (x, y, z, w) = (-1, 3, -1, 4).

Answer:

Step-by-step explanation:

[tex]\frac{(5.27+x)}{2} =-4.51[/tex],[tex]\frac{8.21+y}{2} = 1.37[/tex]

(3.75,-5.47)

Answer:

$26

4/52 = 1/13..  the king will appear one in 13 tries... 13 tries is $26

Step-by-step explanation:

You should expect to spend $26 to win $100 playing this game.

It is the chance of an event to occur from a total number of outcomes.

The formula for probability is given as:

Probability = Number of required events / Total number of outcomes.

Example:

The probability of getting a head in tossing a coin.

P(H) = 1/2

We have,

To calculate the expected cost of playing this game until you win $100, we need to determine the probability of drawing a king on any given turn, as well as the number of times you are expected to play the game before you win.

So,

The probability of drawing a king on any given turn is 4/52, or 1/13 since there are 4 kings in a standard deck of 52 cards.

To determine the number of times you are expected to play the game before you win, we can use the geometric distribution, which models the number of trials it takes to achieve success in a sequence of independent trials, where the probability of success remains constant across trials.

The probability of winning on any given trial is 1/13, and the probability of losing is 12/13.

The expected number of trials until the first success (drawing a king) is:

= 1 / (1/13) = 13

This means that on average, you can expect to play the game 13 times before drawing a king and winning the $100 prize.

Now,

Since each game costs $2 to play, the total cost of playing the game 13 times is:

13 x $2 = $26

Therefore,

You should expect to spend $26 to win $100 playing this game.

Learn more about probability here:

https://brainly.com/question/14099682

#SPJ2

Answer:

8

Step-by-step explanation:

5 = 40

1 = x

Then we multiply by the rule of crisscrossing

5 x X = 40 x 1

5x = 40 then divide both by 5

X = 8

Answer:

maybe 10000

Step-by-step explanation:

Answer:

9007.35

Step-by-step explanation:

First find the effective rate: .06/12= .005

let x= amount

[tex]x=100\frac{1-(1+.005)^{-12*10}}{.005}\\100*\frac{1-.549632733}{.005}\\9007.345333[/tex]

Answer:

The maximum value of the objective function is 112 when x = 0 and y = 7.

Step-by-step explanation:

Given the constraints:

5x+3y≤37, 3x+5y≤35, x≥0, y≥0

Plotting the above constraints using geogebra online graphing tool, we get the solution to the constraints as:

A(0, 7), B(7.4, 0), C(5, 4) and D(0, 0)

The objective function is given as E =2x+16y, therefore:

At point A(0, 7):  E = 2(0) + 16(7) = 112

At point B(7.4, 0): E = 2(7.4) + 16(0) = 14.8

At point C(5, 4): E = 2(5) + 16(4) = 74

At point D(0, 0): E = 2(0) + 16(0) = 0

Therefore the maximum value of the objective function is at A(0, 7).

The maximum value of the objective function is 112 when x = 0 and y = 7.

Answer:

(-5,2)

Step-by-step explanation:

A function is a relation where each y-value has a unique x-value. That means that x's can never repeat. Therefore, to solve find the ordered pair that does not have the same x-value as one of the points on the graph. The x-values currently represented on the graph are -4, -2, 2, 3. So, the only coordinate pair that does not repeat an x-value is the last option, (-5, 2).

Answer:

The first equation; x-2y=8

Step-by-step explanation:

Hi there!

We're told that Ty wants to isolate x in one of the equations. To do so in either, he will need to use inverse operations to cancel out values and leave just x remaining on one side of the equation.

In the second equation, he would need to subtract both sides by 6y and then divide both sides by 4 to isolate x. It's a two-step process.

However, in the first equation, he only needs to add 2y to both sides to isolate x.

I hope this helps!

Answer:

using the first equation

cause Being that the first equation has the simplest coefficients (1, -2, for x, and y respectively), it seems logical to use it to develop a definition of one variable in terms of the other

⭕ 17.3

#CARRYONLEARNING

[tex]{hope it helps}}[/tex]

Answer:

-15 -3.45 -3.15 -3.0

Step-by-step explanation:

Answer:

The distance between the campsites was 70 miles.

Step-by-step explanation:

Since two groups were moving from one campsite to another, and the first group traveled the distance in 5 hours while the second group finished in 7 hours, to find the distance between the campsites if the first group was going 4mph faster than the second group, the following calculation must be performed:

X + 4 = 5

X = 7

4 x 7 = 28

(4 + 4) x 5 = 40

10 x 7 = 70

(10 + 4) x 5 = 70

Therefore, the distance between the campsites was 70 miles.

Answer:

[tex]\frac{8}{45}[/tex]

Step-by-step explanation:

'of' means 'multiply'

4/5 × 2/9 = 8/45

Answer:

300%

Step-by-step explanation:

because 9/3×100=900/3=300 so it is 300%

Answer:

300%

Step-by-step explanation:

9/3 * 100%

900%/3 = 300%

Answer:

1.581

Step-by-step explanation:

Given the data:

13 15 14 16 12

The point estimate of the standard deviation will be :

√Σ(x - mean)²/n-1

Mean = Σx / n = 70 / 5 = 14

√[(13 - 14)² + (15 - 14)² + (14 - 14)² + (16 - 14)² + (12 - 14)² / (5 - 1)]

The point estimate of standard deviation is :

1.581

Answer:

7π/4 radians = 315°, Quadrant IV

sin(315°) = -√2/2

cos(315°) = √2/2

tan(315°) = -1

Step-by-step explanation: