Question

This week a factory is producing 50 units of a particular commodity, and the amount being

produced is increasing at the rate of 2 units per week. If 𝐶(𝑥) dollars is the total cost of producing

𝑥 units and 𝐶(𝑥) = 0.08𝑥^3 − 𝑥^2 + 10𝑥 + 4 , find the current rate at which the production cost is increasing.

Answer 1

$C(x) = 0.08x^3 - x^2 + 10x + 48\\ \frac{\mathrm{d}C}{\mathrm{d}x} = 0.24x^2 - 2x + 10\\ \begin{aligned} \frac{\mathrm{d}C}{\mathrm{d}x}\biggr\vert_{x = 2} &= 0.24(2)^2 - 2(2) + 10 \\&= 0.24\times 4 - 4 + 10 \\&= 0.96 - 4 + 10 = 6.96 \end{aligned}\\ \textsf{Therefore, the current rate at}\\ \textsf{which the production cost}\\ \textsf{is increasing is}\, \, 6.96 \\ \textsf{dollars per units.}$