A cone of height $10\text{cm}$ and radius $4\text{cm}$ is being filled with water at a rate of $2{\text{cm}}^3{\text{s}}^{-1}.$

Initially, the cone was empty.

1. Show that the radius of the surface of the water after $5$ seconds is equal to

   $${\left(\frac{12}{\pi }\right)}^{\frac{1}{3}}$$

2. Find the rate of change of the radius of the surface of the water after $5$ seconds.

3. Find the rate of change of the height of the water after $5$ seconds.

Use similar triangles to show that

Use the first hint to show that

After $5$ seconds, you know that $10{\text{cm}}^3$ of water has been added, so you can find $r.$

Find $\frac{dV}{dr}$ from

You know that $\frac{dV}{dt}$ is $2{\text{cm}}^3{\text{s}}^{-1}$ so you can use

To find $\frac{dh}{dt},$ remember the first hint!

Give

to two decimal places.