Sure, let's solve this step-by-step.
1. Find the Radius of the Base:
- The diameter of the base is given as 8 cm.
- Radius, [tex]\( r \)[/tex], is half of the diameter.
[tex]\[
r = \frac{8}{2} = 4 \text{ cm}
\][/tex]
2. Determine the Vertical Height:
- The vertical height of the cone is three times the diameter.
- Given diameter [tex]\( d \)[/tex] is 8 cm.
[tex]\[
\text{Vertical Height, } h = 3 \times 8 = 24 \text{ cm}
\][/tex]
3. Find the Slant Height:
- We use the Pythagorean theorem in the context of the cone's right triangle (with height [tex]\( h \)[/tex] and radius [tex]\( r \)[/tex]).
[tex]\[
l = \sqrt{h^2 + r^2}
\][/tex]
Substituting our values:
[tex]\[
l = \sqrt{24^2 + 4^2} = \sqrt{576 + 16} = \sqrt{592} \approx 24.33 \text{ cm}
\][/tex]
4. Calculate the Total Surface Area of the Cone:
- The total surface area, [tex]\( A \)[/tex], consists of the base area and the lateral surface area.
- The base area [tex]\( A_{\text{base}} \)[/tex] is given by the formula:
[tex]\[
A_{\text{base}} = \pi r^2
\][/tex]
- The lateral surface area [tex]\( A_{\text{lateral}} \)[/tex] is given by the formula:
[tex]\[
A_{\text{lateral}} = \pi r l
\][/tex]
- Therefore, the total surface area [tex]\( A \)[/tex] is:
[tex]\[
A = A_{\text{base}} + A_{\text{lateral}} = \pi r^2 + \pi r l = \pi r (r + l)
\][/tex]
Substituting our values:
[tex]\[
A = \pi \times 4 \times (4 + 24.33) \approx \pi \times 4 \times 28.33 \approx 356.02 \text{ cm}^2
\][/tex]
So, the total surface area of the cone is approximately 356.02 cm².
