1.12: Rates of change in a triangular prism

Figure 3 shows a metal solid $\displaystyle S$. The solid is a right triangular prism.

The cross section of $\displaystyle S$ is an equilateral triangle with sides of length $\displaystyle x\,\mathrm{cm}$. The length of $\displaystyle S$ is $\displaystyle 4x\,\mathrm{cm}$.

The prism is being heated so that the cross-sectional area is increasing at a constant rate of $\displaystyle 0.03\,\mathrm{cm}^2\mathrm{s}^{-1}$.

(a) Find, giving your answer to 3 significant figures, $\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}$ when $\displaystyle x=2$.

(b) Find the rate of increase, in $\displaystyle \mathrm{cm}^3\mathrm{s}^{-1}$, of the volume of $\displaystyle S$ when $\displaystyle x=2$.