To find the surface area of a right cylinder given the diameter of 28 inches and the height of 9 inches, follow these steps:
1. Calculate the radius of the cylinder:
The radius [tex]\( r \)[/tex] is half of the diameter [tex]\( d \)[/tex].
[tex]\[
r = \frac{d}{2} = \frac{28}{2} = 14 \text{ inches}
\][/tex]
2. Calculate the lateral surface area of the cylinder:
The formula for the lateral surface area of a cylinder is [tex]\( 2\pi rh \)[/tex].
[tex]\[
\text{Lateral Area} = 2\pi rh = 2\pi (14)(9)
\][/tex]
Plugging in the values:
[tex]\[
\text{Lateral Area} = 2 \times \pi \times 14 \times 9 = 791.68 \text{ square inches} \quad (\text{approx.})
\][/tex]
3. Calculate the area of one base:
The formula for the area of a circle is [tex]\( \pi r^2 \)[/tex].
[tex]\[
\text{Base Area} = \pi r^2 = \pi (14)^2
\][/tex]
Plugging in the values:
[tex]\[
\text{Base Area} = \pi \times 14^2 = 615.75 \text{ square inches} \quad (\text{approx.})
\][/tex]
4. Calculate the total surface area:
The total surface area of the cylinder includes the lateral surface area and the area of the two bases.
[tex]\[
\text{Total Surface Area} = \text{Lateral Area} + 2 \times \text{Base Area}
\][/tex]
Plugging in the previously calculated values:
[tex]\[
\text{Total Surface Area} = 791.68 + 2 \times 615.75 = 791.68 + 1231.50 = 2023.18 \text{ square inches} \quad (\text{approx.})
\][/tex]
5. Round the total surface area to the nearest hundredth:
The final result is:
[tex]\[
2023.19 \text{ square inches}
\][/tex]
Therefore, the surface area of the right cylinder with a diameter of 28 inches and a height of 9 inches, rounded to the nearest hundredth, is [tex]\(\boxed{2023.19 \text{ square inches}}\)[/tex].
