Question

question: the volume of a cube increases at a rate of 5 m³/sec. find the rate at which the side of the cube changes when its length is 8 m. recall that the volume of a cube with side of length s is given by v = s³. provide your answer below: $\frac{ds}{dt}=square$ m/sec

Explanation:

Step1: Differentiate volume formula

The volume of a cube is $V = s^{3}$. Differentiating with respect to time $t$ using the chain - rule gives $\frac{dV}{dt}=3s^{2}\frac{ds}{dt}$.

Step2: Substitute known values

We know that $\frac{dV}{dt}=5$ m³/sec and $s = 8$ m. Substitute these values into the equation $5=3\times8^{2}\times\frac{ds}{dt}$.

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

First, calculate $3\times8^{2}=3\times64 = 192$. Then the equation becomes $5 = 192\times\frac{ds}{dt}$. So, $\frac{ds}{dt}=\frac{5}{192}$ m/sec.

Answer:

$\frac{5}{192}$ m/sec