Question

A heap of rubbish in the shape of a cube is being compacted into a smaller cube. Given that the volume decreases at a rate of 7 cubic meters per minute find the rate of change of an edge in meter per minute of the cube when the volume is exactly 27 cubic meters

Answer 1

Ok so we know that the volume of a cube is given as:\[\bf V=s^3\]Where \(\bf V\) is volume and \(\bf s\) is side length. We also know that \(\bf \frac{dV}{dt}=-7 \ cu.m/s\). Let's now differentiate the Volume and plug this value in:\[\bf \frac{ d V}{ dt }=3s^2 \times \frac{ds}{dt}\]We want to evaluate ds/dt when the volume is 27 cu.m. If the volume is 27 cu. m then the side length of the cube at that time must be 3m:\[\bf 27m^3=s^3 \implies s=3m\]Now plug in s = 3m and dV/dt = -7 cu.m/s in to the second equation and solve for ds/dt:\[\bf -7m^3/s=3(3m)^2 \times \frac{ds}{dt} \implies \frac{ds}{dt}=\frac{ -7m^3 }{ s } \times \frac{ 1 }{ 27m^2 }=-\frac{ 7 }{ 27 }m/s\]

Answer 2

ok that makes sense thank you