Certainly! Let's work through the given problem step by step:
### Part (a): Calculating the Height of the Whole Cone
1. Given Data:
- Height of the frustrum, [tex]\( h_{\text{frustrum}} = 12 \)[/tex] cm
- Bottom diameter = 3 * Top diameter
2. Derive the Radii:
- Let the top diameter be [tex]\( d_t \)[/tex]. Therefore, the bottom diameter is [tex]\( 3 \cdot d_t \)[/tex].
- Top radius, [tex]\( r_t = \frac{d_t}{2} \)[/tex].
- Bottom radius, [tex]\( r_b = \frac{3 \cdot d_t}{2} \)[/tex].
Since the top diameter is given as 1 (arbitrary unit for calculations):
- [tex]\( r_t = \frac{1}{2} \)[/tex] units
- [tex]\( r_b = \frac{3}{2} \)[/tex] units
3. Using Similar Triangles To Find Heights:
- Height of the whole cone includes the height of the frustrum and the height of the remaining (small) cone from the top to the top of the frustrum.
- Let the height of the whole cone be [tex]\( H \)[/tex] and the height of the small cone be [tex]\( h_{\text{small}} \)[/tex].
Using the similarity of triangles:
[tex]\[
\frac{h_{\text{small}}}{r_t} = \frac{H}{r_b}
\][/tex]
Solving for [tex]\( H \)[/tex]:
[tex]\[
h_{\text{small}} = H - h_{\text{frustrum}}
\][/tex]
[tex]\[
\frac{H - h_{\text{frustrum}}}{r_t} = \frac{H}{r_b}
\][/tex]
[tex]\[
\frac{H - 12}{\frac{1}{2}} = \frac{H}{\frac{3}{2}}
\][/tex]
[tex]\[
2(H - 12) = \frac{2H}{3}
\][/tex]
[tex]\[
6(H - 12) = 2H
\][/tex]
[tex]\[
6H - 72 = 2H
\][/tex]
[tex]\[
4H = 72
\][/tex]
[tex]\[
H = 18 \text{ cm}
\][/tex]
So, the height of the whole cone is [tex]\( 18 \)[/tex] cm.
### Part (b): Finding the Radius with the Given Volume
1. Given Data:
- Volume of the whole cone, [tex]\( V = 39,600 \)[/tex] cm[tex]\(^3\)[/tex]
- Height of the whole cone, [tex]\( H = 18 \)[/tex] cm
- [tex]\(\pi = \frac{22}{7}\)[/tex]
2. Volume Formula for a Cone:
[tex]\[
V = \frac{1}{3} \pi r^2 H
\][/tex]
3. Solve For Radius [tex]\( r \)[/tex]:
[tex]\[
39,600 = \frac{1}{3} \cdot \frac{22}{7} \cdot r^2 \cdot 18
\][/tex]
[tex]\[
39,600 = \frac{22}{7} \cdot 6 \cdot r^2
\][/tex]
[tex]\[
39,600 = \frac{132}{7} \cdot r^2
\][/tex]
[tex]\[
39,600 \cdot 7 = 132 \cdot r^2
\][/tex]
[tex]\[
277,200 = 132 \cdot r^2
\][/tex]
[tex]\[
r^2 = \frac{277,200}{132}
\][/tex]
[tex]\[
r^2 = 2,100
\][/tex]
[tex]\[
r = \sqrt{2,100}
\][/tex]
[tex]\[
r \approx 12.8058 \text{ cm}
\][/tex]
So, the radius of the base of the whole cone is approximately [tex]\( 12.8058 \)[/tex] cm when rounded to four significant figures.
