Question

Solve the following example: If the total cost function is given by; C = 5x³ + 2x² + 7; find the average cost and the marginal cost when x = 4.

Answer 1

Total cost function, C = 5x³ + 2x² + 7

Average cost = ${\displaystyle \frac{\text{C}}{x}}$

 =${\displaystyle \frac{5x^3+2x^2+7}{x}}$

= 5x² + 2x + ${\displaystyle \frac{7}{x}}$

When x = 4,

Average cost = 5(4)² + 2(4) + ${\displaystyle \frac{7}{4}}$

= 80 + 8 + ${\displaystyle \frac{7}{4}}$

= ${\displaystyle \frac{320+32+7}{4}}$

= ${\displaystyle \frac{359}{4}}$

Marginal cost = ${\displaystyle \frac{\text{dC}}{\text{d}x}}$

=${\displaystyle \frac{\text{d}}{\text{d}x}(5x^3+2x^2+7)}$

= ${\displaystyle 5\frac{\text{d}}{\text{d}x}\left(x^3\right)+2\frac{\text{d}}{\text{d}x}\left(x^2\right)+\frac{\text{d}}{\text{d}x}\left(7\right)}$

= 5(3x²) + 2(2x) + 0
= 15x² + 4x
When x = 4, Marginal cost = ${\displaystyle {\left(\frac{\text{dC}}{\text{dx}}\right)}_{x=4}}$
= 15(4)² + 4(4)
= 240 + 16
= 256
∴ the average cost and marginal cost at x = 4 are ${\displaystyle \frac{359}{4}}$ and 256 respectively.