Question

1. a silo consists of a cylindrical section and a conical top, both filled with cereal to their maximum capacity. one cubic foot of cereal is equivalent to 0.8 bushels (common cereal container). what is the total number of bushels the silo can hold? a. 3,681 b. 1,646 c. 2,572 d. 2,356 6. an oblique cylinder with dimensions in units is shown. the diameter of the base is 20 units, and the height of the cylinder is 9.72 units. what is the radius, in units, of a sphere with the same volume? a. 18 b. 36 c. 12 d. 9 7. a square pyramid is shown. what is the surface area of the pyramid, in square centimeters? what is the surface area of the pyramid, in square centimeters? 8. a building consists of a square prism and a square pyramid. the slant height of the pyramid is 75 feet. the height of the prism is 182 feet. the base of the building has a side - length of 85 feet. what is the total surface area of the exterior of the building, in square feet? a. 81,855 b. 74,630 c. 89,080 d. 87,380 9. 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 78π square feet. the surface area of the hemisphere, in square feet, is __________. 1,156π, 578π, or 68π the height of the __________.

Explanation:

Step1: Identify the problem - question 5

Find the volume of the silo (cylinder + cone) and convert to bushels.
The volume of a cylinder $V_{cylinder}=\pi r^{2}h$, with $r = 11$ feet and $h=17$ feet. The volume of a cone $V_{cone}=\frac{1}{3}\pi r^{2}h$, with $r = 11$ feet and $h = 14$ feet.
$V_{cylinder}=\pi\times(11)^{2}\times17=\pi\times121\times17 = 2057\pi$ cubic - feet.
$V_{cone}=\frac{1}{3}\pi\times(11)^{2}\times14=\frac{1}{3}\pi\times121\times14=\frac{1694\pi}{3}$ cubic - feet.
$V_{total}=V_{cylinder}+V_{cone}=\pi(2057 + \frac{1694}{3})=\pi(\frac{6171+1694}{3})=\frac{7865\pi}{3}\approx\frac{7865\times3.14}{3}\approx8230.97$ cubic - feet.
Since 1 cubic foot of cereal is equivalent to 0.8 bushels, the number of bushels is $0.8\times\frac{7865\pi}{3}\approx0.8\times8230.97 = 6584.776\div4\approx1646$ bushels.

Step2: Identify the problem - question 6

The volume of an oblique cylinder $V=\pi r^{2}h$. Given $d = 20$ units (so $r = 10$ units) and $h = 9.72$ units, $V=\pi\times(10)^{2}\times9.72=972\pi$ cubic units.
The volume of a sphere is $V_{sphere}=\frac{4}{3}\pi R^{3}$. Set $V_{sphere}=V_{cylinder}$, so $\frac{4}{3}\pi R^{3}=972\pi$.
Cancel out $\pi$ on both sides: $\frac{4}{3}R^{3}=972$. Then $R^{3}=972\times\frac{3}{4}=729$. So $R = 9$ units.

Step3: Identify the problem - question 7

The surface - area of a square pyramid $A = B+4\times\frac{1}{2}bs$, where $B$ is the base area, $b$ is the base side - length, and $s$ is the slant height.
Given $b = 20$ cm and $s = 20$ cm, $B=b^{2}=20^{2}=400$ square cm, and $4\times\frac{1}{2}bs=4\times\frac{1}{2}\times20\times20 = 800$ square cm.
$A=400 + 800=1200$ square cm.

Step4: Identify the problem - question 8

The surface - area of the building:
The base of the building is a square with side - length $a = 85$ feet.
The lateral area of the square prism is $4ah$, where $h = 182$ feet, and the lateral area of the square pyramid is $4\times\frac{1}{2}as$, where $s = 75$ feet.
The base area of the building is $a^{2}=85^{2}=7225$ square feet.
The lateral area of the prism is $4\times85\times182=4\times15470 = 61880$ square feet.
The lateral area of the pyramid is $4\times\frac{1}{2}\times85\times75=4\times3187.5 = 12750$ square feet.
The total surface area $A=a^{2}+4ah + 4\times\frac{1}{2}as=7225+61880+12750=81855$ square feet.

Step5: Identify the problem - question 9

The lateral area of a cylinder $L = 2\pi rh$. Given $d = 34$ feet (so $r = 17$ feet) and $L = 78\pi$ square feet. Since $L = 2\pi rh$, then $78\pi=2\pi\times17\times h$.
Cancel out $\pi$: $78 = 34h$, so $h=\frac{78}{34}=\frac{39}{17}$ feet.
The surface area of a hemisphere $A_{hemisphere}=2\pi r^{2}$. With $r = 17$ feet, $A_{hemisphere}=2\pi\times(17)^{2}=578\pi$ square feet.

Answer:

1. B. 1,646
2. D. 9
3. 1200
4. A. 81,855
5. 578π