Answer :
We are given a cone with a diameter of 3 inches, so the radius is
[tex]$$
r = \frac{3}{2} = 1.5 \text{ inches.}
$$[/tex]
The volume of a cone is given by
[tex]$$
V = \frac{1}{3} \pi r^2 h.
$$[/tex]
Here, the volume is 12 cubic inches. Substituting the known values into the volume formula gives:
[tex]$$
12 = \frac{1}{3} \pi (1.5)^2 h.
$$[/tex]
First, calculate [tex]$(1.5)^2$[/tex]:
[tex]$$
(1.5)^2 = 2.25.
$$[/tex]
So the equation becomes:
[tex]$$
12 = \frac{1}{3} \pi \cdot 2.25 \cdot h.
$$[/tex]
To isolate [tex]$h$[/tex], multiply both sides by 3:
[tex]$$
36 = \pi \cdot 2.25 \cdot h.
$$[/tex]
Now, solve for [tex]$h$[/tex] by dividing both sides by [tex]$\pi \cdot 2.25$[/tex]:
[tex]$$
h = \frac{36}{\pi \cdot 2.25}.
$$[/tex]
Evaluating the right-hand side yields approximately:
[tex]$$
h \approx 5.092958 \text{ inches.}
$$[/tex]
Rounding to the nearest inch gives:
[tex]$$
h \approx 5 \text{ inches.}
$$[/tex]
Thus, the height of the cone is [tex]$\boxed{5}$[/tex] inches.
[tex]$$
r = \frac{3}{2} = 1.5 \text{ inches.}
$$[/tex]
The volume of a cone is given by
[tex]$$
V = \frac{1}{3} \pi r^2 h.
$$[/tex]
Here, the volume is 12 cubic inches. Substituting the known values into the volume formula gives:
[tex]$$
12 = \frac{1}{3} \pi (1.5)^2 h.
$$[/tex]
First, calculate [tex]$(1.5)^2$[/tex]:
[tex]$$
(1.5)^2 = 2.25.
$$[/tex]
So the equation becomes:
[tex]$$
12 = \frac{1}{3} \pi \cdot 2.25 \cdot h.
$$[/tex]
To isolate [tex]$h$[/tex], multiply both sides by 3:
[tex]$$
36 = \pi \cdot 2.25 \cdot h.
$$[/tex]
Now, solve for [tex]$h$[/tex] by dividing both sides by [tex]$\pi \cdot 2.25$[/tex]:
[tex]$$
h = \frac{36}{\pi \cdot 2.25}.
$$[/tex]
Evaluating the right-hand side yields approximately:
[tex]$$
h \approx 5.092958 \text{ inches.}
$$[/tex]
Rounding to the nearest inch gives:
[tex]$$
h \approx 5 \text{ inches.}
$$[/tex]
Thus, the height of the cone is [tex]$\boxed{5}$[/tex] inches.
