Question

How to answer this ?

Answer 1

The height is changing at approximately `0.239` centimetres per minute.

Explanation:

Recall that the volume of a cone is given by `V = 1/3r^2hpi`. However, since we have no information about the radius, we must eliminate the radius parameter using trigonometry.

Since the angle between the surface and the height is 90˚, we can say that `tan(45˚) = r/h`, thus `1 = r/h` and `r = h`. Therefore, the volume in this case is given by `V = 1/3h^2hpi = 1/3h^3pi`

The derivative with respect to time for this is obtained easily using the power rule.

`(dV)/(dt) = h^2pi(dH)/(dt)`

We know the rate of change of volume and we know the water level. All we need to solve for is `(dH)/(dt)`.

`3 = 2^2 pi(dH)/(dt)`

`3/(4pi) = (dH)/(dt)`

`(dH)/(dt) ~~ 0.239`

Hopefully this helps!