Question

1. a museum is designing a new exhibit about space composed of a right cylinder and a hemisphere, as shown. the diameter of the base of the cylinder is 34 feet and the lateral area of the cylinder is square feet. the surface area of the hemisphere, in square feet, is _. 1,156π, 578π, or 68π the height of the cylinder, in feet, is _. 22, 34, or 44

Explanation:

Step1: Find radius of hemisphere

The diameter of the base of the cylinder (which is equal to the diameter of the hemisphere) is $d = 34$ feet. The radius $r=\frac{d}{2}=\frac{34}{2}=17$ feet.

Step2: Calculate surface - area of hemisphere

The formula for the surface area of a hemisphere is $2\pi r^{2}$. Substitute $r = 17$ into the formula: $2\pi(17)^{2}=2\pi\times289 = 578\pi$ square feet.

Step3: Recall formula for lateral area of cylinder

The formula for the lateral area of a cylinder is $L = 2\pi rh$. We know that $L$ (lateral area of cylinder) and $r$. Given $L = 2\pi rh$ and $r = 17$, and assume $L= 2344\pi$ (not shown in the problem - statement but needed for the next step. If we assume the lateral area is $2344\pi$). Then $2\pi rh=2344\pi$. Divide both sides by $2\pi r$: $h=\frac{2344\pi}{2\pi r}$. Substitute $r = 17$ into the formula: $h=\frac{2344\pi}{2\pi\times17}=\frac{2344}{34}= 69$ (This is wrong. Let's assume the lateral area is $2344\pi$ for illustration. If we assume the correct lateral - area value based on the multiple - choice values for height). If we use the formula $L = 2\pi rh$, and we know $r = 17$. If we try $h = 22$, $L=2\pi\times17\times22=2\pi\times374 = 748\pi$. If $h = 34$, $L=2\pi\times17\times34=2\pi\times578 = 1156\pi$. If $h = 44$, $L=2\pi\times17\times44=2\pi\times748 = 1496\pi$. Let's assume the lateral area is $1156\pi$. Then $h = 34$.

Answer:

The surface area of the hemisphere is $578\pi$.
The height of the cylinder is $34$.