Optimization Application

Question

The mass of a cube in grams is x^3 + 0.1x^4, where  is the length of one side in centimeters. If the length is increasing at a rate of 0.02 cm/hr, at what rate is the mass of the cube increasing when its length is  7cm? The mass of the cube is increasing at a rate of (fill in the blank) grams per hour.

Answer 1

M = x³ + 0.1x⁴

Let M by the mass of the cube, let x = the length of a side of the cube, let dx/dt be the rate at which the side length is changing (which we are given as 0.02), and let dM/dt be the rate at which the mass is changing.

Take the derivative with respect to time:

dM/dt = 3x²•dx/dt + 0.4x³•dx/dt

dM/dt = dx/dt(3x² + 0.4x³)

dM/dt = (0.02)(3(7²) + 0.04(7³)) now just do the arithmetic.