The question aims to find the volume of the given cone with 14 inches in height and 10 inches in diameter as shown in Figure 1.
The question depends on the cone geometry. A cone is a 3D solid shape with a circular body at the bottom and a triangular shape towards the top. The volume of the cone can be calculated by the formula given below:
\[ Volume\ V = \dfrac{ 1 }{ 3 } \pi r^2 h \]
Here,
\[ r = Radius\ of\ the\ circular\ body\ of\ the\ cone \]
\[ h = Height\ of\ the\ cone \]
Expert Answer
The given information about the question is as follows:
\[ Diameter\ d = 10\ in \]
\[ Height\ h = 14\ in \]
To find the radius of the cone, we divide the diameter in half to calculate its radius, which is given as:
\[ r = \dfrac{ d }{ 2 } \]
\[ r = \dfrac{ 10 }{ 2 } \]
\[ r = 5\ in \]
The formula for the volume of the cone is given as:
\[ Volume\ V = \dfrac{ 1 }{ 3 } \pi r^2 h \]
Substituting the values, we get:
\[ V = \dfrac{ 1 }{ 3 } \pi \times (5)^2 \times 14 \]
\[ V = \dfrac{ 1 }{ 3 } \pi \times 25 \times 14 \]
\[ V = \dfrac{ 1 }{ 3 } 350 \pi \]
\[ V = 116.67 \pi \]
\[ V = 366.52 {in}^3 \]
Numerical Result
The volume of the cone with 10 inches in diameter and 14 inches in height is calculated to be:
\[ V = 366.52 {in}^3 \]
Example
Find the volume of the given cone below with 12 inches in diameter and 10 inches in height.
Figure 2
\[ Height\ h = 10\ in \]
\[ Diameter\ d = 12\ in \]
In order to calculate the radius of the cone, we divide the diameter in half to calculate its radius, which is given as:
\[ r = \dfrac{ d }{ 2 } \]
\[ r = \dfrac{ 12 }{ 2 } \]
\[ r = 6\ in \]
The formula for the volume of the cone is given as:
\[ Volume\ V = \dfrac{ 1 }{ 3 } \pi r^2 h \]
Substituting the values, we get:
\[ V = \dfrac{ 1 }{ 3 } \pi \times (6)^2 \times 10 \]
\[ V = \dfrac{ 1 }{ 3 } \pi \times 36 \times 10 \]
\[ V = \dfrac{ 1 }{ 3 } 360 \pi \]
\[ V = 120\pi \]
\[ V = 377 {in}^3 \]
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