Question

A company distributes college logo sweatshirts and sells them for $50 each. The total cost function is linear, and the total cost for 100 sweatshirts is $4796, whereas the total cost for 240 sweatshirts is $7036.

(a) Write the equation for the revenue function R(x).


R(x) =




(b) Write the equation for the total cost function C(x).


C(x) =




(c) Find the break-even quantity.


x =


sweatshirts

Answer 1

• R(x) = 50x
• C(x) = 16x +3196
• x = 94

Explanation:

(a) The revenue is simply $50 per shirt, where x is the number of shirts sold:

R(x) = 50x

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(b) We are given two points on the linear cost curve, so we can write the equation using the 2-point form of the equation of a line.

y = (y2 -y1)/(x2 -x1)(x -x1) +y1

The two points we are given are (100, 4796) and (240, 7036). Then the cost equation is ...

C(x) = (7036 -4796)/(240 -100)(x -100) +4796

= (2240/140)(x -100) +4796

= 16x -1600 +4796

C(x) = 16x +3196

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(c) The break-even quantity is the value of x that makes revenue equal to cost.

R(x) = C(x)

50x = 16x +3196

34x = 3196 . . . . . . . . subtract 16x

x = 94 . . . . . . . . . . . . divide by 34

The break-even quantity is 94 sweatshirts.