To determine the height of a cone given its diameter and volume, follow these steps:
1. Identify the given values:
- Diameter of the cone: 3 inches
- Volume of the cone: 12 cubic inches
2. Determine the radius of the cone:
- The radius is half the diameter.
- So, radius [tex]\( r = \frac{diameter}{2} = \frac{3}{2} = 1.5 \)[/tex] inches.
3. Recall the formula for the volume of a cone:
- The volume [tex]\( V \)[/tex] of a cone is given by [tex]\( V = \frac{1}{3} \pi r^2 h \)[/tex], where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height.
- We need to solve for the height [tex]\( h \)[/tex].
4. Rearrange the formula to solve for the height [tex]\( h \)[/tex]:
- [tex]\( V = \frac{1}{3} \pi r^2 h \)[/tex]
- Multiply both sides by 3 to isolate the term with [tex]\( h \)[/tex]: [tex]\( 3V = \pi r^2 h \)[/tex]
- Divide both sides by [tex]\( \pi r^2 \)[/tex] to solve for [tex]\( h \)[/tex]: [tex]\( h = \frac{3V}{\pi r^2} \)[/tex]
5. Substitute the known values into the formula:
- [tex]\( V = 12 \)[/tex] cubic inches
- [tex]\( r = 1.5 \)[/tex] inches
- So, [tex]\( h = \frac{3 \times 12}{\pi \times (1.5)^2} \)[/tex]
6. Calculate the height:
- First, calculate [tex]\( (1.5)^2 = 2.25 \)[/tex]
- Then, calculate [tex]\( \pi \times 2.25 \)[/tex]
- Finally, [tex]\( h = \frac{36}{\pi \times 2.25} \)[/tex]
7. Evaluating the expression:
- [tex]\( h \approx 5.092958178940651 \)[/tex] inches
8. Round the height to the nearest inch:
- [tex]\( h \approx 5 \)[/tex] inches
Given the result of evaluating and rounding, the height of the cone to the nearest inch is [tex]\( \boxed{5} \)[/tex].
So the correct answer is:
C. 5
