Question

Each side of a square is increasing at a rate of 4 cm/s. At what rate (in cm2/s) is the area of the square increasing when the area of the square is 25 cm2

Answer 1

The area of the square is increasing at a rate of 40 square centimeters per second.

Explanation:

The area of the square (
`A`), in square centimeters, is represented by the following function:

`A = l^(2)` (1)

Where
`l` is the side length, in centimeters.

Then, we derive (1) in time to calculate the rate of change of the area of the square (
`(dA)/(dt)`), in square centimeters per second:

`(dA)/(dt) = 2\cdot l \cdot (dl)/(dt)`

`(dA)/(dt) = 2\cdot √(A)\cdot (dl)/(dt)` (2)

Where
`(dl)/(dt)` is the rate of change of the side length, in centimeters per second.

If we know that
`A = 25\,cm^(2)` and
`(dl)/(dt) = 4\,(cm)/(s)`, then the rate of change of the area of the square is:

`(dA)/(dt) = 2\cdot \sqrt{25\,cm^(2)}\cdot \left(4\,(cm)/(s) \right)`

`(dA)/(dt) = 40\,(cm^(2))/(s)`

The area of the square is increasing at a rate of 40 square centimeters per second.