Let’s solve the given problem step by step.

a) Determine the radius of the circles:

The rectangle $ABCD$ has dimensions 40 (height) and 60 (width). The circles $E$ and $F$ are tangent to three sides of the rectangle.

1. Observation: Since the circles are tangent to the top, bottom, and one vertical side of the rectangle, their radius $r$ is equal to half the height of the rectangle.

   From the height: $r = \frac{\text{height of the rectangle}}{2} = \frac{40}{2} = 20$

Thus, the radius $r$ of each circle is 20.

b) Calculate the area of the shaded region:

1. Shaded region: The shaded region is the intersection of the two circles, which forms a "lens" shape.

2. Centers of the circles: The centers $E$ and $F$ are positioned along the length of the rectangle. The distance between them is the width of the rectangle minus the diameter of one circle: $d = \text{width} - 2r = 60 - 40 = 20$

3. Formula for the intersection area of two circles: For two circles of radius $r$ whose centers are separated by distance $d$, the area of their intersection is given by: $A = 2r^2 \cos^{-1}\left(\frac{d}{2r}\right) - \frac{d}{2} \sqrt{4r^2 - d^2}$ Substituting $r = 20$ and $d = 20$: $A = 2(20)^2 \cos^{-1}\left(\frac{20}{2(20)}\right) - \frac{20}{2} \sqrt{4(20)^2 - (20)^2}$

4. Simplify terms:

   • $\cos^{-1}\left(\frac{20}{40}\right) = \cos^{-1}(0.5) = \frac{\pi}{3}$
   • $\sqrt{4(20)^2 - (20)^2} = \sqrt{1600 - 400} = \sqrt{1200} = 20\sqrt{3}$ Substituting: $A = 2(400) \cdot \frac{\pi}{3} - 10 \cdot 20\sqrt{3}$ $A = \frac{800\pi}{3} - 200\sqrt{3}$

Thus, the area of the shaded region is: $\boxed{\frac{800\pi}{3} - 200\sqrt{3}}$

Summary:

• (a) The radius of the circles is $\mathbf{20}$.
• (b) The area of the shaded region is $\mathbf{\frac{800\pi}{3} - 200\sqrt{3}}$.

Do you want a detailed breakdown of any step, or any further explanation? Here are 5 related questions to explore:

1. How is the formula for the intersection area of two circles derived?
2. What happens to the shaded area if the radius is doubled?
3. Can you visualize the intersection area geometrically without calculus?
4. How does the center distance $d$ influence the shaded area?
5. How would the problem change if the rectangle dimensions were different?

Tip: Always verify the tangency conditions for circles inside geometric shapes to simplify calculations!