Answer :
To find the surface area of a cone, we need to consider both the base area and the lateral surface area. Given the problem:
- Height of the cone (h) = 28 inches
- Diameter of the base (d) = 14 inches
- Radius of the base (r) = diameter / 2 = 14 / 2 = 7 inches
The first step is to determine the slant height (l) of the cone. The slant height can be found using the Pythagorean theorem:
[tex]\[ l = \sqrt{h^2 + r^2} \][/tex]
Substitute the given values:
[tex]\[ l = \sqrt{28^2 + 7^2} \approx 28.86 \text{ inches} \][/tex]
Next, calculate the base area (A_base) of the cone using the formula for the area of a circle:
[tex]\[ A_{\text{base}} = \pi r^2 \][/tex]
[tex]\[ A_{\text{base}} = \pi \times 7^2 \approx 153.94 \text{ square inches} \][/tex]
Then, we calculate the lateral surface area (A_lateral) of the cone using the formula:
[tex]\[ A_{\text{lateral}} = \pi r l \][/tex]
[tex]\[ A_{\text{lateral}} = \pi \times 7 \times 28.86 \approx 634.70 \text{ square inches} \][/tex]
Finally, to find the total surface area (A_total) of the cone, sum the base area and the lateral surface area:
[tex]\[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} \][/tex]
[tex]\[ A_{\text{total}} = 153.94 + 634.70 \approx 788.64 \text{ square inches} \][/tex]
Rounding to the nearest square inch, the surface area is:
[tex]\[ \text{Surface area} \approx 789 \text{ square inches} \][/tex]
Therefore, the correct answer is:
- Option C: 789 in²
- Height of the cone (h) = 28 inches
- Diameter of the base (d) = 14 inches
- Radius of the base (r) = diameter / 2 = 14 / 2 = 7 inches
The first step is to determine the slant height (l) of the cone. The slant height can be found using the Pythagorean theorem:
[tex]\[ l = \sqrt{h^2 + r^2} \][/tex]
Substitute the given values:
[tex]\[ l = \sqrt{28^2 + 7^2} \approx 28.86 \text{ inches} \][/tex]
Next, calculate the base area (A_base) of the cone using the formula for the area of a circle:
[tex]\[ A_{\text{base}} = \pi r^2 \][/tex]
[tex]\[ A_{\text{base}} = \pi \times 7^2 \approx 153.94 \text{ square inches} \][/tex]
Then, we calculate the lateral surface area (A_lateral) of the cone using the formula:
[tex]\[ A_{\text{lateral}} = \pi r l \][/tex]
[tex]\[ A_{\text{lateral}} = \pi \times 7 \times 28.86 \approx 634.70 \text{ square inches} \][/tex]
Finally, to find the total surface area (A_total) of the cone, sum the base area and the lateral surface area:
[tex]\[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} \][/tex]
[tex]\[ A_{\text{total}} = 153.94 + 634.70 \approx 788.64 \text{ square inches} \][/tex]
Rounding to the nearest square inch, the surface area is:
[tex]\[ \text{Surface area} \approx 789 \text{ square inches} \][/tex]
Therefore, the correct answer is:
- Option C: 789 in²
