Related Rates Problem

Question

A square tile is expanding as it is heated. Right now, the 16 cm tile is growing by 1.75 cm per hour. At what rate is the area of the tile changing?



Answer 1

We want to find how the area is changing, which should scream "first derivative." So we want dA/dt, where A(t) is the area of the tile over time. But how can we get A(t)?

In these related rates problems, we want to write everything we know, as if we differentiate, often what we want to find out will pop out. We know A(s) = s² where s is the length of one side as the tile is a square. And we know that ds/dt = 1.75 cm/hr and s(t) = 16 + 1.75t, as the tile starts at 16 cm so s(0) = 16.

Using the chain rule, dA/dt = dA/ds ds/dt = 2s ds/dt = 2s * 1.75. But s(0) = 16, and we can assume the question is asking about the current moment! Letting 1.75 = 7/4 for ease of arithmetic, we have that dA/dt = 2 * 16 * 7/4 = 8 * 7 = 56 cm per hour.