Question

question the volume of a cube decreases at a rate of 10 m³/s. find the rate at which the side of the cube changes when the side of the cube has a length of 2 m. enter a fraction and do not include units. provide your answer below:

Explanation:

Step1: Volume formula of cube

$V = x^{3}$, where $x$ is side - length. Differentiate with respect to time $t$: $\frac{dV}{dt}=3x^{2}\frac{dx}{dt}$.

Step2: Substitute known values

Given $\frac{dV}{dt}=- 10$ (decreasing) and $x = 2$. Then $-10=3\times2^{2}\times\frac{dx}{dt}$.

Step3: Solve for $\frac{dx}{dt}$

$\frac{dx}{dt}=\frac{-10}{3\times4}=-\frac{5}{6}$.

Answer:

$-\frac{5}{6}$