To find the surface area of the paper covering the cone, we need to consider both the lateral (side) surface area and the base area of the cone. We are given the following details about the cone:
- Height = 28 inches
- Diameter = 14 inches
Here is the step-by-step solution:
1. Calculate the radius of the base of the cone:
Since the diameter of the cone is 14 inches, the radius (r) is:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{14}{2} = 7 \text{ inches}
\][/tex]
2. Calculate the slant height (l) of the cone:
The slant height can be found using the Pythagorean theorem in the right triangle formed by the radius, the height, and the slant height:
[tex]\[
\text{Slant Height}, \, l = \sqrt{(\text{Height})^2 + (\text{Radius})^2} = \sqrt{(28)^2 + (7)^2} = \sqrt{784 + 49} = \sqrt{833} \approx 28.86 \text{ inches}
\][/tex]
3. Calculate the lateral surface area of the cone:
The formula for the lateral surface area (L) of a cone is:
[tex]\[
L = \pi \times \text{Radius} \times \text{Slant Height}
\][/tex]
Substituting the known values:
[tex]\[
L = \pi \times 7 \times 28.86 \approx 635 \text{ square inches}
\][/tex]
4. Calculate the base area of the cone:
The area of the base (B) is given by the formula for the area of a circle:
[tex]\[
B = \pi \times (\text{Radius})^2
\][/tex]
Substituting the known values:
[tex]\[
B = \pi \times (7)^2 = \pi \times 49 \approx 154 \text{ square inches}
\][/tex]
5. Calculate the total surface area:
The total surface area (T) is the sum of the lateral surface area and the base area:
[tex]\[
T = L + B = 635 + 154 = 789 \text{ square inches}
\][/tex]
After rounding to the nearest square inch, the correct surface area required to cover the cone is:
789 square inches
Therefore, the correct answer is:
[tex]\[
\boxed{B. \, 789 \, \text{in}^2}
\][/tex]
