To determine the surface area of the capsule, we need to break down the problem into the areas of its constituent parts: the cylinder and the two hemispherical ends.
1. Determine the radius of the hemispheres and the cylinder:
- Given the diameter is 0.5 inches, the radius (r) would be:
[tex]\[
r = \frac{0.5}{2} = 0.25 \text{ inches}
\][/tex]
2. Calculate the surface area of the cylindrical part:
- We are given that the height (h) of the cylindrical part is 1 inch.
- The formula for the lateral surface area of a cylinder is:
[tex]\[
A_{\text{cylinder}} = 2 \pi r h
\][/tex]
- Substituting the radius and the height into the formula gives:
[tex]\[
A_{\text{cylinder}} = 2 \pi (0.25)(1) = 0.5 \pi \approx 1.57 \text{ square inches}
\][/tex]
3. Calculate the surface area of the hemispherical ends:
- The formula for the surface area of a sphere is:
[tex]\[
A_{\text{sphere}} = 4 \pi r^2
\][/tex]
- Since we have two hemispheres, together they form a complete sphere. For two hemispheres:
[tex]\[
A_{\text{hemispheres}} = \left(\frac{1}{2} \cdot 4 \pi r^2\right) \times 2 = 2 \pi r^2
\][/tex]
- Substituting the radius:
[tex]\[
A_{\text{hemispheres}} = 2 \pi (0.25)^2 = 2 \pi (0.0625) = 0.125 \pi \approx 0.79 \text{ square inches}
\][/tex]
4. Calculate the total surface area of the capsule by adding the surface areas of the cylinder and the hemispheres:
[tex]\[
A_{\text{total}} = A_{\text{cylinder}} + A_{\text{hemispheres}} = 1.57 + 0.79 = 2.36 \text{ square inches}
\][/tex]
Thus, rounding to the nearest hundredth, the total surface area of the capsule is:
Correct answer:
A. [tex]$2.36 \text{ in}^2$[/tex]
