To calculate the surface area of a right circular cylinder, we need to consider both the lateral surface area and the area of the two circular bases.
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Lateral Surface Area: The lateral surface area (A_lateral) of a cylinder can be calculated using the formula: \[ A_{\text{lateral}} = 2\pi rh \]
Where \( r \) is the radius and \( h \) is the height.
Given:
- Diameter of the cylinder = 5 inches, so the radius \( r = \frac{5}{2} = 2.5 \) inches.
- Height \( h = 4 \) inches.
Now substitute the values: \[ A_{\text{lateral}} = 2 \times 3.14 \times 2.5 \times 4 \] \[ A_{\text{lateral}} = 2 \times 3.14 \times 10 = 62.8 \text{ inches}^2 \]
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Area of the Bases: The area of one circular base (A_base) is given by the formula: \[ A_{\text{base}} = \pi r^2 \]
Therefore, the total area of the two bases is: \[ A_{\text{bases}} = 2 \times \pi r^2 \]
Substituting the radius: \[ A_{\text{bases}} = 2 \times 3.14 \times (2.5)^2 \] \[ A_{\text{bases}} = 2 \times 3.14 \times 6.25 = 39.25 \text{ inches}^2 \]
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Total Surface Area: Now add the lateral area and the area of the two bases: \[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{bases}} \] \[ A_{\text{total}} = 62.8 + 39.25 = 102.05 \text{ inches}^2 \]
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