Find the surface area of a cylinder with a radius of 5 inches and a height of 10 inches. Use the approximation 3.14 for pi.
To find the surface area of a cylinder, we need to calculate the area of the two circular bases and the area of the curved side (the lateral surface).
The formula for the area of a circle is \( A_{\text{circle}} = \pi r^2 \), where \( r \) is the radius. The cylinder has two circular bases:
So the combined area of the two bases is:
\[ 2 \cdot A_{\text{circle}} = 2 \cdot \pi r^2 \]
The lateral surface area of a cylinder (the area of the curved side) can be found by "unrolling" the side into a rectangle. The width of this rectangle is the circumference of the base circle, and the height is the same as the height of the cylinder. The circumference \( C \) is given by \( C = 2\pi r \).
So the lateral surface area \( A_{\text{lateral}} \) is:
\[ A_{\text{lateral}} = C \cdot h = 2\pi r \cdot h \]
Now, we can find the total surface area \( A_{\text{total}} \) by adding the areas of the two bases and the lateral surface:
\[ A_{\text{total}} = 2 \cdot A_{\text{circle}} + A_{\text{lateral}} \]
Let's calculate it step by step using the given approximation of \( \pi \approx 3.14 \), with \( r = 5 \) inches and \( h = 10 \) inches.
Area of the two bases:
\[ 2 \cdot A_{\text{circle}} = 2 \cdot (3.14 \cdot 5^2) = 2 \cdot (3.14 \cdot 25) = 2 \cdot 78.5 = 157 \text{ square inches} \]
Lateral surface area:
\[ A_{\text{lateral}} = 2 \cdot 3.14 \cdot 5 \cdot 10 = 314 \text{ square inches} \]
Finally, add the two areas together to get the total surface area:
\[ A_{\text{total}} = 157 \text{ square inches} + 314 \text{ square inches} = 471 \text{ square inches} \]
Therefore, the surface area of the cylinder with a radius of 5 inches and a height of 10 inches, using \( \pi \approx 3.14 \), is 471 square inches.