Let's solve this step-by-step.
1. Identify the given dimensions:
- Radius (r) = 8 inches
- Height (h) = 15 inches
2. Calculate the slant height (l) of the cone using the Pythagorean theorem:
- The slant height can be calculated using the formula: [tex]\( l = \sqrt{r^2 + h^2} \)[/tex]
- Substituting the given values: [tex]\( l = \sqrt{8^2 + 15^2} \)[/tex]
- So, [tex]\( l = \sqrt{64 + 225} \)[/tex]
- [tex]\( l = \sqrt{289} \)[/tex]
- [tex]\( l = 17 \)[/tex]
3. Calculate the lateral surface area of the cone:
- The formula for the lateral surface area is [tex]\( \pi r l \)[/tex]
- Substituting the given values: [tex]\( \pi \times 8 \times 17 \)[/tex]
4. Calculate the base area of the cone:
- The formula for the base area is [tex]\( \pi r^2 \)[/tex]
- Substituting the given values: [tex]\( \pi \times 8^2 \)[/tex]
- [tex]\( \pi \times 64 \)[/tex]
5. Calculate the total surface area:
- The total surface area of the cone is the sum of the lateral surface area and the base area: [tex]\( \pi r l + \pi r^2 \)[/tex]
- Substituting the given values:
[tex]\[
\pi \times 8 \times 17 + \pi \times 64
\][/tex]
- Simplifying this,
[tex]\[
136\pi + 64\pi
\][/tex]
[tex]\[
200\pi
\][/tex]
6. Convert the surface area to a numerical value and round to the nearest whole number:
- Numeric approximation: [tex]\( 200 \times 3.141592653589793 \approx 628.3185307179587 \)[/tex]
- Rounding 628.3185307179587 to the nearest whole number gives us 628.
Thus, the amount of material used for the hat, rounded to the nearest whole number, is about 628 square inches.
