To find the surface area of a cylindrical drum, we can use the formula for the surface area of a cylinder, which includes the areas of the two circular bases and the lateral surface area.
The formula for the surface area \( A \) of a cylinder is given by:
\[ A = 2\pi r^2 + 2\pi rh \]
where:
- \( r \) is the radius of the base,
- \( h \) is the height of the cylinder.
Step 1: Find the radius
Given the diameter of the drum is 13 inches, we can find the radius by dividing the diameter by 2:
\[ r = \frac{d}{2} = \frac{13 \text{ inches}}{2} = 6.5 \text{ inches} \]
Step 2: Use the height
The height \( h \) of the drum is provided as 19 inches.
Step 3: Calculate the surface area
Now we can substitute the values of \( r \) and \( h \) into the surface area formula.
First, calculate the area of the bases:
\[ 2\pi r^2 = 2\pi (6.5)^2 = 2\pi (42.25) \approx 2 \times 3.14 \times 42.25 \approx 265.58 \text{ square inches} \]
Next, calculate the lateral surface area:
\[ 2\pi rh = 2\pi (6.5)(19) = 2\pi (123.5) \approx 2 \times 3.14 \times 123.5 \approx 776.57 \text{ square inches} \]
Step 4: Total surface area
Now, add both areas together to get the total surface area:
\[ A \approx 265.58 + 776.57 \approx 1042.15 \text{ square inches} \]
Conclusion
Thus, the approximate surface area of Peter's drum is:
\[ \boxed{1042.15} \text{ square inches} \]
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