AMC 12 2022b Problem 13

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$. Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

size(5cm);
filldraw((4,0)--(8,3)--(8-3/4,4)--(1,4)--cycle,mediumgray);
draw((0,0)--(8,0)--(8,4)--(0,4)--cycle,linewidth(1.1));
draw((1,0)--(1,4)--(4,0)--(8,3)--(5,7)--(1,4),linewidth(1.1));
label("$4$", (8,2), E);
label("$8$", (4,0), S);
label("$5$", (3,11/2), NW);
draw((1,.35)--(1.35,.35)--(1.35,0),linewidth(1.1));

$\textbf{(A) }15\dfrac{1}{8} \textbf{(B) }15\dfrac{3}{8} \textbf{(C) }15\dfrac{1}{2} \textbf{(D) }15\dfrac{5}{8} \textbf{(E) }15\dfrac{7}{8}$