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Answer :
To solve the problem of finding the total surface area of a cylinder when the circumference of the base is 22 inches and the sum of the radius and height is 15 inches, follow these steps:
1. Understand the problem:
- You are given the circumference of the base of the cylinder: 22 inches.
- The sum of the radius and the height of the cylinder is 15 inches.
2. Find the radius:
- The formula for the circumference of the base circle is [tex]\( C = 2\pi r \)[/tex].
- Given that the circumference [tex]\( C = 22 \)[/tex], you can find the radius [tex]\( r \)[/tex] using the formula:
[tex]\[
r = \frac{C}{2\pi} = \frac{22}{2\pi} \approx 3.5 \, \text{inches}
\][/tex]
3. Calculate the height:
- The sum of the radius and the height is given as 15 inches. Since the radius is approximately 3.5 inches, the height [tex]\( h \)[/tex] can be calculated as:
[tex]\[
h = 15 - r = 15 - 3.5 = 11.5 \, \text{inches}
\][/tex]
4. Find the curved surface area:
- The curved surface area (lateral surface area) of a cylinder is calculated by the formula:
[tex]\[
\text{Curved Surface Area} = 2\pi rh
\][/tex]
- Substituting the values of [tex]\( r \approx 3.5 \)[/tex] and [tex]\( h \approx 11.5 \)[/tex] into the formula gives:
[tex]\[
\text{Curved Surface Area} \approx 2\pi \times 3.5 \times 11.5 \approx 252.97 \, \text{square inches}
\][/tex]
5. Find the total surface area:
- The total surface area of the cylinder includes the curved surface area and the areas of the top and bottom circles. The formula is:
[tex]\[
\text{Total Surface Area} = 2\pi r^2 + 2\pi rh
\][/tex]
- Calculate the total surface area using [tex]\( r \approx 3.5 \)[/tex] and [tex]\( h \approx 11.5 \)[/tex]:
[tex]\[
\text{Total Surface Area} \approx 2\pi \times 3.5^2 + 2\pi \times 3.5 \times 11.5 \approx 330 \, \text{square inches}
\][/tex]
Thus, the curved surface area of the cylinder is approximately 252.97 square inches, and the total surface area is approximately 330 square inches.
1. Understand the problem:
- You are given the circumference of the base of the cylinder: 22 inches.
- The sum of the radius and the height of the cylinder is 15 inches.
2. Find the radius:
- The formula for the circumference of the base circle is [tex]\( C = 2\pi r \)[/tex].
- Given that the circumference [tex]\( C = 22 \)[/tex], you can find the radius [tex]\( r \)[/tex] using the formula:
[tex]\[
r = \frac{C}{2\pi} = \frac{22}{2\pi} \approx 3.5 \, \text{inches}
\][/tex]
3. Calculate the height:
- The sum of the radius and the height is given as 15 inches. Since the radius is approximately 3.5 inches, the height [tex]\( h \)[/tex] can be calculated as:
[tex]\[
h = 15 - r = 15 - 3.5 = 11.5 \, \text{inches}
\][/tex]
4. Find the curved surface area:
- The curved surface area (lateral surface area) of a cylinder is calculated by the formula:
[tex]\[
\text{Curved Surface Area} = 2\pi rh
\][/tex]
- Substituting the values of [tex]\( r \approx 3.5 \)[/tex] and [tex]\( h \approx 11.5 \)[/tex] into the formula gives:
[tex]\[
\text{Curved Surface Area} \approx 2\pi \times 3.5 \times 11.5 \approx 252.97 \, \text{square inches}
\][/tex]
5. Find the total surface area:
- The total surface area of the cylinder includes the curved surface area and the areas of the top and bottom circles. The formula is:
[tex]\[
\text{Total Surface Area} = 2\pi r^2 + 2\pi rh
\][/tex]
- Calculate the total surface area using [tex]\( r \approx 3.5 \)[/tex] and [tex]\( h \approx 11.5 \)[/tex]:
[tex]\[
\text{Total Surface Area} \approx 2\pi \times 3.5^2 + 2\pi \times 3.5 \times 11.5 \approx 330 \, \text{square inches}
\][/tex]
Thus, the curved surface area of the cylinder is approximately 252.97 square inches, and the total surface area is approximately 330 square inches.
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