Question

A firm has a demand function p=108-5q and the cost function c=-12q+q^2. find the price at which profit is maximum. find the maximum profit. also find the output for maximum cost.
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Answer 1

Answer:

$(a)\ q=3.5$

$(b)\ t_{max} = 120.25$

(c) No maximum

Explanation:

Given

$p = 108 - 5q$

$c = -12q + q^2$

Solving (a): The price at maximum profit.

The profit (t) is calculated using the following function

$t = p - c$ --- price- cost

So, we have:

$t = 108 - 5q - (-12q + q^2)$

Open bracket

$t = 108 - 5q +12q - q^2$

$t = 108 +7q - q^2$

Rewrite as:

$t = - q^2 + 7q + 108$ --- this is the profit function

A quadratic function is represented as:

$y = ax^2 + bx + c$

The maximum is:

$x = -\frac{b}{2a}$

By comparison:

$q = -\frac{7}{2*-1}$

$q=3.5$ ----- price at maximum profit

Solving (b): The maximum profit

The profit function is:

$t = - q^2 + 7q + 108$

The maximum is:

$t_{max} = -3.5^2 + 7 * 3.5 +108$

$t_{max} = 120.25$

Solving (c): Maximum cost

We have:

$c = -12q + q^2$

The maximum price of the above function is:

$q = -\frac{-12}{2*1}$ ---- from x = -b/2a

$q = \frac{12}{2}$

$q = 6$

So, the maximum cost is:

$c_{max} = -12 * 6 + 6^2$

$c_{max} = -36$

The cost function has no maximum; only minimum