Question

A hypothetical square shrinks at a rate of 2 m²/min. At what rate are the diagonals of the square changing when the diagonals are 7 m each?

Answer 1

The diagonal is given by

`d = sqrt(s^2 + s^2) = sqrt(2s^2)`, or `s = sqrt(d^2/2) = d/sqrt(2)`.

Since the area of a square is given by `A = s^2`, we have:

`A = (d/sqrt(2))^2 = d^2/2`

Differentiating:

`(dA)/dt= (4d(dd)/(dt))/4`

`(dA)/dt = d(dd)/(dt)`

We know that the area changes with respect to time at `2" m^2"` per minute and we want to find at which rate the diagonals are changing when `d = 7" meters"`.

`-2 = 7(dd)/(dt)`

`-2/7 = (dd)/(dt)`

So, the diagonals shrink at a rate of `-2/7` metres per minute.

Hopefully this helps!