Question

28) Economics A student club is making t-shirts to sell. It cost $250 for the silk-screen design, plus $3.50 to press each shirt. The shirts are purchased for $4.00 each and they decide to sell them for $12.00 each. (a) Write the total cost, revenue, and profit functions. . (b) Find the number of shirts that need to be sold to break even. (c) Find the average rate of change of the profit function for a production level of 10 to 100 shirts.

Answer 1

a)

Let Q be the number of shirts produced.

Total cost = Fixed cost + Variable cost + Purchasing cost

Where fixed cost = Cost for silk screen design ; Variable cost = Cost to press each shirt & Purchasing cost = purchasing cost for each shirt

Total cost = $ 250 + $ 3.50 Q + $ 4.00 Q

Total cost = $ 250 + $ 7.50 Q

Revenue = Price * Quantity

Revenue = $ 12 Q

Profit = Revenue - Total cost

Profit = $ 12 Q - [ $ 250 + $ 7.50 Q ]

Profit = $ 4.5 Q - $ 250

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b)

Fixed cost Break - even = Sel Selling price - variable cost

$250 Break-even number of shirts = $12.00 - $3.50

Break-even number of shirts $\approx$ 30 units

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c)

-f(a) Average rate of change - f b-a

When Q = 100,

Profit = $ 4.5 x 100 units - $ 250

Profit = $ 200

When Q = 10

Profit = $ 4.5 x 10 - $ 250

Profit = - $ 205

$200 - (-$205) Average rate of change = - 100 - 10

Average rate of change of profit function for a production level of 10 to 100 units = $ 4.50