Take Home Quiz #13 MATH 1324 T11 &T17; 2. A screen printer is selling T-shirts.

Question

Take Home Quiz #13 MATH 1324 T11 &T17; 2. A screen printer is selling T-shirts. Past experience indicates that if the price is set at $15 per shirt, they will sell 600 shirts. For each $1 increase in price, they will sell 50 fewer shirts. (a) find the demand function px (b) find the revenue function R(x) assuming a linear relationship. (c) Find the price per shirt and number of shirts sold to maximize revenue. Fixed costs for producing the shirts are $250 and production costs are 5.50 per shirt (d) Find the cost function C(x) (e) Find the profit function P(x). (f) Find the break even quantity (ies) and interpret. (g) Find the price per shirt and number of shirts sold to maximize profit.

Explanation / Answer

(a)

let p(x) = no. of tshirts sold

x = price

so p = mx + c

m = slope = -50 (as with each increment of $1, p fall by 50)

p = -50x+c

put (15,600)

600 = -50*15 + c

c = 1350

p(x) = -50x + 1350

(b)

R(x) = p(x) * x = -50x2+1350x

(c)

R(x) = p(x) * x = -50x2+1350x

first derivative

R'(x) = -100x + 1350

x = $13.5

maximum revenue R(13.5) =  -50*(13.5)2+1350*13.5 = $9,112.5

(d)

cost = 5.5 p(x) + 250

(e)

Profit P(x) = R(x) - C(x) = -50x2+1350x - 5.5 (-50x + 1350) + 250 = -50x2+1625x - 7175

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