How do I find the change in area?

Question

The length of a rectangle is increasing at a rate of

3 meters per day and the width is decreasing at a rate

of 1 meter per day.

When the length is 10 meters and the width is 18 meters,

then how fast is the AREA changing?

The rate of change of AREA is  square meters per day.

(Enter your answer as an integer.)

Answer 1

The equation for the area is area(length(t), width(t)) = length(t) * width(t)

Writing length and width as l, w, we can use the product rule: and get:

(d area) / dt = (dw / dt) * l(t) + (dl / dt) * w(t)

From the problem we have:

dw / dt = -1 meters /day

dl / dt = 3 meters / day

w(t) = 18 meters

l(t) = 10 meters

d area / dt = -1 meters / day * 10 meters + 3 meters / days * 18 meters

Multiplying we get:

d area / dt = -10 meters^2 / day + 54 meters^2 / day = 44 meters^2 / day