Question

A hypothetical square grows at a rate of 16 m²/min. How fast are the sides of the square increasing when the sides are 15 m each?

Answer 1

The sides are increasing at a speed of `8/15` meters/minute.

Explanation:

The formula for area of a square is `A = s^2`, where `s` is the side length.

Differentiating `A` with respect to time:

`(dA)/dt = 2s((ds)/dt)`

Solve for `(ds)/dt`, since this represents the change in the sides with respect to time.

`((dA)/dt)/(2s) = (ds)/dt`

`1/(2s) xx (dA)/dt = (ds)/dt`

Here's what we know and what would be our unknown:

-We know the speed at which the area is changing (16 `m^2`/min)
-We want to know the speed at which the lengths of our sides are changing at the moment when the sides are `15` meters each.

`1/(2 xx 15) xx 16 =(ds)/dt`

`1/30 xx 16 =( ds)/dt`

`8/15 = (ds)/dt`

Hence, the length of the sides are increasing at a speed of `8/15 "m"/"min"` when the sides are at length `15` meters.

Hopefully this helps!