Answer
$w=\frac{3}{2}$, $h=2$
Work Step by Step
Step 1. Set up the coordinate system as shown in the figure in the book, with the bottom left corner labeled as A, the bottom right corner labeled as C, and the top corner labeled as B.
Step 2. We can identify the coordinates of the points as $A(0,0)$ and $B(3,4)$ and we can find the line equation of AB as $y=\frac{4}{3}x$
Step 3. The upper left corner $(3-w,h)$ of the rectangle is on the hypotenuse AB; thus $h=\frac{4}{3}(3-w)$
Step 4. The area of the rectangle is $A=wh=\frac{4}{3}(3w-w^2)$. Take the derivative to get $A'=\frac{4}{3}(3-2w)$ and letting $A'=0$, we get $w=\frac{3}{2}$, which gives $h=\frac{4}{3}(3-\frac{3}{2})=2$ and $A=3unit^2$.
Step 5. Check $A''=-\frac{8}{3}\lt0$ and we can confirm that the region is concave down with a local maximum at the above point.
