You are an auto shop supervisor in charge of assembling cars and painting cars. You earn $250 for every car assembled and $150 for every car painted. It takes 3 hours to assemble a car and you need 3 workers to do it. It takes 2 hours to paint a car but you need 4 workers to accomplish it. You only have 24 hours to work on this task and 36 employees at work today. Write, graph, and solve a linear programming system and find how many cars can be assembled and painted of each you need to make in order to maximize your earnings. State the maximum earnings made.

The number of cars that can be assembled in order to make maximum earnings is 6 cars

The number of cars that can be assembled in order to make maximum earnings is 4 cars

The maximum earnings = $2,100

Please find attached the graph of the system of equations

Step-by-step explanation:

The given parameters are;

The amount earned for assembling a car = $250

The number of hours it takes to assemble a car = 3 hours

The number of workers needed to assemble a car = 3 workers

The amount earned for painting a car = $150

The number of hours it takes to paint a car = 2 hours

The number of workers needed to paint a car = 4 workers

The total number of hours available = 24 hours

The number of employees at work = 36 employees

Let the number cars assembled = X

The number of cars painted = Y

Therefore, we have;

The linear programming system is therefore;

3X + 4Y ≤ 36

3X + 2Y ≤ 24

X ≥ 0

Y ≥ 0

Y ≤ (36 - 3X)/4

Y ≤ (24 - 3X)/2

Graphing the function gives;

The amount earned for each car assembled = $250

The amount earned for each car painted = $150

Earnings, E = 250X + 150Y

Y = (-250/150)·X + E/150 = (-5/3)·X + E/150

Y = (-5/3)·X + E/150

Therefore the slope of the maximum earnings line is -5/3

Drawing several lines to find one that both passes through the feasible region and gives the highest y-intercept, gives the line that passes through the point (4, 6)

6 = (-5/3)·4 + E/150

From which we have E = 38/3.

Therefore, the auto shop supervisor should assemble 6 cars and paint 4 to earn E = $250 × 6 + $150 × 4 = $2,100.

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

step-by-step explanation:

a and d