A company distributes college logo sweatshirts and sells them for $55 each. The total cost function is linear, and the total cost for 50 sweatshirts is $4346, whereas the total cost for 240 sweatshirts is $7576. (a) Write the equation for the revenue function R(x).

the revenue function is R(Q)=$55/swshrt * Q and the profit function is P(Q)=$3496 - $38/swshrt * Q

Explanation:

since the cost function is linear , then denoting C as cost , Q as quantities of sweatshirts and 1 and 2 as reference points , we get

C= C₁ + (C₂-C₁)/(Q₂-Q₁)*(Q-Q₁)

replacing values

C= $4346 + ($7576- $4346)/(240 -50 )*(Q-50 )=  $4346 + $17/swshrt*(Q- 50)

and the revenue function (total sales is )

R= P*Q = $55/swshrt * Q

the profit function is therefore

P = R - C =  $55/swshrt * Q - [$4346 + $17/swshrt*(Q- 50)] =  $3496 - $38/swshrt * Q

Notes

- Revenue refers to the total income generated , while profit refers to the income after costs and expenses

- We can verify the cost equation for C . For Q=Q₁

C= C₁ + (C₂-C₁)/(Q₂-Q₁)*(Q₁-Q₁) = C₁ + 0 =  C₁

and for Q=Q₂

C= C₁ + (C₂-C₁)/(Q₂-Q₁)*(Q₂-Q₁) = C₁ + C₂-C₁ = C₂

thus our equation is correct

The revenue equation is TR=55*P

Total cost equation is TC=3496+17Q

Explanation:

The formula for revenue is given quantity multiplied by  price

TR=PQ

P is the price per unit ]

Q is the total quantity

since P is $55

TR=55P

The total cost function can also be determined thus:

TC =a+bQ

where a is the fixed cost

b is variable cost per unit

Q is the number of output

when TC is $4346,Q was 50

when TC was $7576 Q was 240

7576=a+240b

4346=a+50b

subtracting the two equations

3230 =190b

b=3230/190

b=$17

by substituting b in any of the equations,fixed cost can be determined

4346=a+(17*50)

4346=a+850

a=4346-850

a=3496

Total cost function

TC=3496+17Q