Burger Bun

The formula for the curved surface area of a spherical cap is $2 \pi r h$ , where $r$ is the radius of the sphere from which the cap is cut, and $h$ is the height measured from the center of the base of the cap to its apex. We know the heights of the two caps are $1$ and $2$ ; we need to find their respective radii.

Let $r$ and $R$ be the radii of the spheres from which the top and bottom caps respectively were cut, as shown below.

$\begin{aligned} (r - 1)^2 &+ 4^2 = r^2 \qquad \qquad \qquad \qquad \qquad \qquad \qquad & (R - 2)^2 &+ 4^2 = R^2 \\ \\ \text{-} 2r + &17 = 0 \qquad \qquad \qquad \qquad \qquad \qquad \qquad & \text{-} 4R + &20 = 0 \\ \\ r =& \dfrac{17}{2} \qquad \qquad \qquad \qquad \qquad \qquad \qquad & R &= 5 \end{aligned}$

Then the combined curved surface area of the two caps is $2 \pi \left( \dfrac{17}{2} \right) (1) + 2 \pi (5) (2) = \boxed{37 \pi}$

The area of a spherical zone is given by $z=2\pi R h$ where: $R=radius~of~a~great~circle$ and $h=height$ . There is a derived formula for the radius of a great circle. Applying this formula , the radius of the upper spherical segment is $\dfrac{17}{2}$ and the lower spherical segment is $5$ . Applying the formula for the area of a spherical zone, the area of the upper zone is $17\pi$ and the lower zone is $20\pi$ . So the area of the bun is $17\pi + 20\pi = 37 \pi$ . The answer we are looking for is $\boxed{37}$ .

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Note: A spherical zone is that portion of the surface of a sphere included between two parallel bases. In this problem, there is only one base.