Solve the problem.
-A dome is in the form of a partial sphere, with a hemisphere of radius 10 feet on top and the remaining part of the sphere extending 5 feet to the ground from the center of the sphere. Find the surface area of the dome to the nearest square foot.
A) $9 \mathrm { ft } ^ { 2 }$
B) $9425 \mathrm { ft } ^ { 2 }$
C) $94 \mathrm { ft } ^ { 2 }$
D) $942 \mathrm { ft } ^ { 2 }$

Question

Quizplus · Accepted Answer

Since we only need to find the surface area of the dome, we can break it down into two parts - the hemisphere on top and the remaining part of the sphere below.

The hemisphere on top has a surface area of $2\pi r^2$, where $r$ is the radius of the hemisphere, so in this case it is $2\pi(10)^2=200\pi$ square feet.

For the remaining part of the sphere, we need to find the surface area of the entire sphere and then subtract the surface area of the hemisphere. The radius of the entire sphere is $10+5=15$ feet, so the surface area of the entire sphere is $4\pi(15)^2=900\pi$ square feet. The surface area of the hemisphere is already included in the surface area of the entire sphere, so we need to subtract it once to get the surface area of the remaining part of the sphere. The surface area of the hemisphere is $\frac{1}{2}(200\pi)=100\pi$ square feet, so the surface area of the remaining part of the sphere is $900\pi - 100\pi = 800\pi$ square feet.

Adding the surface area of the hemisphere and the remaining part of the sphere together, we get a total surface area of $200\pi + 800\pi = 1000\pi$ square feet. To the nearest square foot, this is approximately $942$ square feet, so our answer is D.

Solve the Problem 9ft29 \mathrm { ft } ^ { 2 }9ft2

Solve the Problem $9 \mathrm { ft } ^ { 2 }$