Answer
$A(h)=\sqrt{100-h^2}\cdot h$
Work Step by Step
We know that the area of a rectangle is base times height $A=bh$, but we are asked to represent this function only in terms of height. Therefore, we must find a relation between $b$ and $h$ in this situation.
We can see that if we trace an imaginary line from the center of the semicircle to one of the intersections of the corner of the rectangle and the semicircle, a right triangle will be formed. With that, we can use the Pythagoras's Theorem to find the relation using the radius $(10)$ as the hypotenuse, the base as one of its legs, and the height as the other leg:
$10^2=b^2+h^2$ so, we'll solve for $b$
$10^2-h^2=b^2+h^2-h^2$
$b^2=10^2-h^2$
$b=\sqrt{100-h^2}$
Now, we substitute it into the area's equation to make it in terms of $h$ only:
$A(h)=\sqrt{100-h^2}\cdot h$
