To find the surface area of a capsule with a given diameter of 0.5 inches, follow these steps:
1. Calculate the radius from the diameter:
The diameter of the hemispheres is 0.5 inches. The radius (r) is half of the diameter:
[tex]\[
r = \frac{0.5}{2} = 0.25 \text{ inches}
\][/tex]
2. Calculate the surface area of the sphere formed by the two hemispheres:
The total surface area of a sphere is given by the formula:
[tex]\[
\text{Surface Area of Sphere} = 4 \pi r^2
\][/tex]
Plug in the radius:
[tex]\[
\text{Surface area of sphere} = 4 \pi (0.25)^2 = 4 \pi (0.0625) \approx 0.7853981633974483 \text{ square inches}
\][/tex]
3. Calculate the surface area of the cylindrical part:
The surface area of a cylinder (excluding the top and bottom circles) is given by the formula:
[tex]\[
\text{Surface Area of Cylinder} = 2 \pi r h
\][/tex]
Here, we assume the height (h) of the cylinder is equal to the diameter of the hemispheres, which is 0.5 inches:
[tex]\[
\text{Surface area of cylinder} = 2 \pi (0.25) (0.5) = 2 \pi (0.125) \approx 0.7853981633974483 \text{ square inches}
\][/tex]
4. Calculate the total surface area of the capsule:
Sum the surface area of the spherical part and the cylindrical part:
[tex]\[
\text{Total Surface Area} = \text{Surface area of sphere} + \text{Surface area of cylinder}
\][/tex]
[tex]\[
\text{Total Surface Area} = 0.7853981633974483 + 0.7853981633974483 = 1.5707963267948966 \text{ square inches}
\][/tex]
5. Round the total surface area to the nearest hundredth:
[tex]\[
1.5707963267948966 \approx 1.57 \text{ square inches}
\][/tex]
Therefore, the surface area of the capsule, rounded to the nearest hundredth, is [tex]\(1.57 \text{ in}^2\)[/tex]. The correct answer is:
C. [tex]\(1.57 \text{ in}^2\)[/tex]
