Sure, let's find the volume of a frustum of a cone with the given dimensions.
The formula to calculate the volume [tex]\( V \)[/tex] of a frustum of a cone is:
[tex]\[ V = \frac{1}{3} \pi h (r_1^2 + r_2^2 + r_1 \cdot r_2) \][/tex]
where:
- [tex]\( r_1 \)[/tex] is the radius of the smaller base
- [tex]\( r_2 \)[/tex] is the radius of the larger base
- [tex]\( h \)[/tex] is the height of the frustum
- [tex]\( \pi \)[/tex] is a constant approximately equal to 3.14159
Given:
- [tex]\( r_1 = 3 \)[/tex] inches
- [tex]\( r_2 = 10 \)[/tex] inches
- [tex]\( h = 10 \)[/tex] inches
Let's plug these values into the formula:
1. First, evaluate the squared radii and the product of the radii:
[tex]\[ r_1^2 = 3^2 = 9 \][/tex]
[tex]\[ r_2^2 = 10^2 = 100 \][/tex]
[tex]\[ r_1 \cdot r_2 = 3 \cdot 10 = 30 \][/tex]
2. Add these terms together:
[tex]\[ r_1^2 + r_2^2 + r_1 \cdot r_2 = 9 + 100 + 30 = 139 \][/tex]
3. Multiply by the height [tex]\( h \)[/tex] and [tex]\(\pi \)[/tex]:
[tex]\[ \pi h (r_1^2 + r_2^2 + r_1 \cdot r_2) = \pi \cdot 10 \cdot 139 \][/tex]
4. Divide by 3 to complete the formula:
[tex]\[ V = \frac{1}{3} \pi \cdot 10 \cdot 139 \approx 1455.60 \][/tex] cubic inches
Therefore, the volume of the frustum of the cone is approximately [tex]\( 1455.60 \)[/tex] cubic inches.
Out of the provided options, the closest value to our result is:
- 1454.86 in³
So, the correct answer is:
1454.86 in³
