Question

A company manufactures soccer balls. Your profit, P(n), can be modeled by the function P(n)= -0.003n^2 + 5.4n - 1211. Where n is the number of soccer balls sold each month.

a.) how many soccer balls must you make and sell in order to break even?

b.) How many soccer balls must you make and sell in order to maximize profit?

c.) What is the maximum profit?

d.) How many soccer balls must you make and sell in order to earn a profit of $677?

Answer 1

Ans. Profit function, P = -0.003n^2 + 5.4n - 1211

a) To break even means earning zero profits,

=> P = 0

=> -0.003n^2 + 5.4n - 1211 = 0

=> n = 263.56 or 1537.44units

Thus, 1537 soccer balls (this is because increasing output from 263 to 1537 does not reduced the profit) should be made in order to break even.

b) To maximize the profit, differentiate the profit fuction with respect to n and then equate it to zero,

=> dP/dn = -0.006n + 5.4 = 0

=> n = 900 units

So, 900 soccer balls should be made to maximize profits.

c) Maximum profit, P = -0.003*900^2 + 5.4*900 -1211

=> Maximum Profit = $1219

d) To earn a profit of $677,

677 = -0.003*n^2 +5.4n - 1211

=> n = 474.95 or 1325.05 units

So, to earn a profit of $677, the production will be 1325 units (because after 1325.05 profits start decreasing)

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