A capsule is made with a rectangular solid base that has dimensions of 14.4 cm × 14.4 cm × 18.6 cm. The cone sits on top and has a slant height of 23.6 cm. Determine the volume of the capsule.

A) $4,880.05 \mathrm {~cm} ^ { 3 }$
B) $5,076.98 \mathrm {~cm} ^ { 3 }$
C) $8,737.25 \mathrm {~cm} ^ { 3 }$
D) Insufficient information

Question

A capsule is made with a rectangular solid base that has dimensions of 14.4 cm × 14.4 cm × 18.6 cm. The cone sits on top and has a slant height of 23.6 cm. Determine the volume of the capsule.
 
A) $4,880.05 \mathrm {~cm} ^ { 3 }$ 
B) $5,076.98 \mathrm {~cm} ^ { 3 }$ 
C) $8,737.25 \mathrm {~cm} ^ { 3 }$ 
D) Insufficient information

Quizplus · Accepted Answer

The volume of the rectangular solid base is:

$V_{1} = l\times w\times h = 14.4\times 14.4\times 18.6 = 3,912.96~\text{cm}^{3}$

The formula for the volume of a cone is:

$V_{2} = \dfrac{1}{3}\pi r^2 h$

where $r$ is the radius of the base and $h$ is the height of the cone.

Since the slant height is given, we can use the Pythagorean theorem to find the radius:

$r=\sqrt{h^2+(\frac{w}{2})^2}=\sqrt{23.6^2+(\frac{14.4}{2})^2}=24.8\text{ cm}$

Thus, the volume of the cone is:

$V_{2} = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3}\pi \times 24.8^2 \times 5 = 5,164.02~\text{cm}^3$

The total volume of the capsule is the sum of the volumes of the rectangular solid base and the cone:

$V = V_{1} + V_{2} = 3,912.96~\text{cm}^{3}+5,164.02~\text{cm}^3=9,077.98~\text{cm}^3$

Thus, the best choice is B, which gives the volume as approximately $5,076.98 \mathrm {~cm} ^ { 3 }$.

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A Capsule Is Made with a Rectangular Solid Base That $4,880.05 \mathrm {~cm} ^ { 3 }$