Question

1. find the volume of the right - circular cone given the dimensions r = 6 ft, h = 1/3 ft. find the volume of the sphere in terms of π and to the nearest cubic foot.
2. a sphere has a radius of 10 feet. find the volume of the sphere in terms of π and to the nearest cubic foot.
3. find the volume of the following figure: a figure with dimensions 8 in, 8 in, 12 in is shown
4. the height of the pyramid is 4 inches, and the base is a rectangle 6 inches long and 3 1/2 inches wide. find the volume.
5. a square pyramid has a volume of 81 ft³. if the height of the pyramid is 9 ft, find the area of the base.

Explanation:

Step1: Recall volume formula for cone

The volume formula for a right - circular cone is $V=\frac{1}{3}\pi r^{2}h$. Given $r = 6$ ft and $h=\frac{1}{3}$ ft. Substitute the values into the formula: $V=\frac{1}{3}\pi(6)^{2}\times\frac{1}{3}$.

Step2: Calculate the volume

First, $(6)^{2}=36$. Then $\frac{1}{3}\pi\times36\times\frac{1}{3}=\frac{36\pi}{9} = 4\pi$ cubic feet. To the nearest cubic foot, $V\approx13$ cubic feet.

Step3: Recall volume formula for sphere

The volume formula for a sphere is $V=\frac{4}{3}\pi r^{3}$. Given $r = 10$ feet. Substitute $r = 10$ into the formula: $V=\frac{4}{3}\pi(10)^{3}$.

Step4: Calculate the volume of the sphere

$(10)^{3}=1000$, so $V=\frac{4000\pi}{3}$ cubic feet. To the nearest cubic foot, $V\approx4189$ cubic feet.

Step5: Analyze the composite figure

The figure is a combination of a rectangular - based pyramid and a rectangular prism. The volume of the rectangular prism is $V_{prism}=l\times w\times h=8\times8\times8 = 512$ cubic inches. The volume of the rectangular - based pyramid on top has base length $l = 8$ in, base width $w = 8$ in and height $h=12 - 8=4$ in. The volume formula for a rectangular - based pyramid is $V_{pyramid}=\frac{1}{3}lwh$. So $V_{pyramid}=\frac{1}{3}\times8\times8\times4=\frac{256}{3}$ cubic inches.

Step6: Calculate the total volume of the composite figure

$V = V_{prism}+V_{pyramid}=512+\frac{256}{3}=\frac{1536 + 256}{3}=\frac{1792}{3}\approx597$ cubic inches.

Step7: Recall volume formula for rectangular - based pyramid

The volume formula for a rectangular - based pyramid is $V=\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height. The base is a rectangle with length $l = 6$ inches and width $w = 3.5$ inches, so $B=lw=6\times3.5 = 21$ square inches. Given $h = 4$ inches. Then $V=\frac{1}{3}\times21\times4=28$ cubic inches.

Step8: Recall volume formula for square - based pyramid

The volume formula for a square - based pyramid is $V=\frac{1}{3}Bh$, where $B$ is the area of the base and $h$ is the height. Given $V = 81$ cubic feet and $h = 9$ feet. We can solve for $B$ from the formula $B=\frac{3V}{h}$.

Step9: Calculate the area of the base

Substitute $V = 81$ and $h = 9$ into the formula: $B=\frac{3\times81}{9}=27$ square feet.

Answer:

1. $4\pi$ ft³, approximately 13 ft³
2. $\frac{4000\pi}{3}$ ft³, approximately 4189 ft³
3. $\frac{1792}{3}$ in³, approximately 597 in³
4. 28 in³
5. 27 ft²