Question

a manufacturer knows that if x hundred of products are demanded in a particular week his total cost function would be TC=14+3x and the coreesponding revenue function TR+19x-2x2
derive the profit function
find the breakeven point
calculate the level of demand that maximizes the profit of the company and hence calculate the maximum profits

Answer 1

Answer to Question #88266 - Math - Calculus

Question

a manufacturer knows that if x hundred of products are demanded in a particular week his total cost function would be TC=14+3x and the corresponding revenue function TR+19x-2x2

derive the profit function

find the breakeven point

calculate the level of demand that maximizes the profit of the company and hence calculate the maximum profits

Solution

Cost function

$$C.F = 14 + 3x$$

Revenue function

$$R.F = 19x - 2x^2$$

Thus Profit function

P.F=Revenue Function-Cost function

$$P.F = 19x - 2x^2 - 14 - 3x$$

For breakeven point

Revenue function=cost function

$$\begin{array}{l} 19x - 2x^2 = 14 + 3x \\ 2x^2 - 16x + 14 = 0 \\ 2x(x-1) - 14(x-1) = 0 \\ (x-1)(2x-14) = 0 \\ x-1 = 0 \text{ or } 2x-14 = 0 \\ \end{array}$$

x=1 or 2x=14

x=1 or x=7

we get

x=1; 7

Both value of x are the break even points

Now compute the maximum value of profit

Set derivative of profit function to be equal to zero

Thus 19-4x-3=0

We get x=4

The maximum value of profit is

$$(19 \times 4) - (2 \times 4^2) - 14 - (3 \times 4)$$

=18

at x=4.

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