To calculate the surface area of a right circular cylinder, we can use the following formula:
\[ \text{Surface Area} = 2\pi r(h + r) \]
where \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
From the information you provided:
- The height of the cylinder (h) is 4 inches.
- The diameter of the cylinder is 5 inches, which means the radius (r) is half of the diameter, so \( r = \frac{5}{2} = 2.5 \) inches.
Now, we can plug these values into the formula:
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Calculate \( h + r \): \[ h + r = 4 + 2.5 = 6.5 \text{ inches} \]
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Calculate the surface area: \[ \text{Surface Area} = 2 \cdot 3.14 \cdot 2.5 \cdot 6.5 \]
Calculating \( 2 \cdot 3.14 \cdot 2.5 \): \[ 2 \cdot 3.14 = 6.28 \] \[ 6.28 \cdot 2.5 = 15.7 \]
Now, calculate \( 15.7 \cdot 6.5 \): \[ 15.7 \cdot 6.5 = 102.05 \]
So, the surface area is approximately \( 102.05 \) square inches, which does not match any of the answers given in your list.
Let’s check the full component for surface area: The surface area of a cylinder is actually calculated as: \[ \text{Surface Area} = 2\pi r^2 + 2\pi rh \] If we substitute:
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The area of the two circular bases: \[ 2\pi r^2 = 2 \cdot 3.14 \cdot (2.5)^2 \] \[ (2.5)^2 = 6.25 \] \[ 2 \cdot 3.14 \cdot 6.25 = 39.25 \text{ square inches} \]
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The lateral surface area: \[ 2\pi rh = 2 \cdot 3.14 \cdot 2.5 \cdot 4 = 2 \cdot 3.14 \cdot 10 \] \[ 2 \cdot 3.14 = 6.28 \] \[ 6.28 \cdot 10 = 62.8 \text{ square inches} \]
Now combine both areas: \[ \text{Total Surface Area} = 39.25 + 62.8 = 102.05 \text{ square inches} \text{ (as calculated)} \]
It seems there was confusion with the dimensions or the options provided are incorrect. However, based on the given measurements, the calculation is valid, and the surface area is approximately 102.05 square inches.
None of the options seem correct; there might be a misalignment in the problem or values provided. Please ensure the dimensions and the context align with the expected answers.