To solve this problem, we will apply the formulas for the properties of a cone step-by-step using the provided values.
### Given:
- Diameter of the cone's base, [tex]\( d = 60 \)[/tex] cm
- Volume of the cone, [tex]\( V = 12,000 \pi \)[/tex] cm³
### Steps:
#### 1. Calculate the radius of the base:
The radius [tex]\( r \)[/tex] can be found using the diameter:
[tex]\[ r = \frac{d}{2} = \frac{60}{2} = 30 \text{ cm} \][/tex]
#### 2. Calculate the height of the cone:
The volume [tex]\( V \)[/tex] of a cone is given by the formula:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
We can rearrange this formula to solve for the height [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3V}{\pi r^2} \][/tex]
Substitute [tex]\( V \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ h = \frac{3 \times 12,000 \pi}{\pi \times 30^2} = \frac{36,000 \pi}{900 \pi} = \frac{36,000}{900} = 40 \text{ cm} \][/tex]
#### 3. Calculate the slant height of the cone:
The slant height [tex]\( l \)[/tex] can be found using the Pythagorean theorem in the right triangle formed by the radius, the height, and the slant height:
[tex]\[ l = \sqrt{r^2 + h^2} \][/tex]
Substitute [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ l = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \text{ cm} \][/tex]
#### 4. Calculate the total surface area of the cone:
The total surface area [tex]\( TSA \)[/tex] of a cone is given by the formula:
[tex]\[ TSA = \pi r (r + l) \][/tex]
Substitute [tex]\( r \)[/tex] and [tex]\( l \)[/tex]:
[tex]\[ TSA = \pi \times 30 \times (30 + 50) = \pi \times 30 \times 80 = 2400 \pi \text{ cm}^2 \][/tex]
### Final Answers:
(i) Height:
[tex]\[ h = 40 \text{ cm} \][/tex]
(ii) Slant height:
[tex]\[ l = 50 \text{ cm} \][/tex]
(iii) Total surface area:
[tex]\[ TSA = 2400 \pi \text{ cm}^2 \approx 7539.82 \text{ cm}^2 \][/tex]
