Question

At a certain point in time, each dimension of a cube of ice is 30 cm and is decreasing at the rate of 2 cm/hr. How fast is the ice melting (losing volume)?

Answer 1

Recall that for a cube, all dimensions, length, width and height, are the same in value.

so, if say length = width = height = x, then the volume of the cube is V = x³, therefore

$\bf V=x^3\implies \cfrac{dV}{dt}=\stackrel{chain~rule}{3x^2\cfrac{dx}{dt}}\quad \begin{cases} x=30\\ \frac{dx}{dt}=\stackrel{cm/hr}{2} \end{cases}\implies \cfrac{dV}{dt}=3(30)^2(2) \\\\\\ \cfrac{dV}{dt}=5400$