An ant is on top of an apple and wants to crawl down. The apple is modeled as a sphere of radius $5\text{cm}.$ The ant is trying to reach the ground as quickly as possible and therefore its path forms a semi-circle.

The ant's height decreases at a constant rate of $0.5\text{cm}{\text{ s}}^{-1}$ (it slows down as it gets tired).

1. Find the horizontal displacement of the ant $4$ seconds after it begins its path.

2. Find the horizontal speed of the ant at this time.

What is the equation of the circle?

You know that the height will have decreased by $2\text{cm}$ when $t=4,$ so you can find the $y$ coordinate of $P$.

You have

Use this to find $\frac{dy}{dx}$ (implicit differentiation is the easiest way, but rearranging then using the chain rule will also work).

What you want to find is $\frac{dx}{dt}.$

Careful! Because $y$ is decreasing we have

Use this along with the previous hint and the coordinates of $P$ to find $\frac{dx}{dt},$ this is the horizontal speed.

Give your answer as a decimal