To find the surface area of the capsule, we need to consider the surface area of both the cylindrical part and the two hemispheres at either end. Here's a step-by-step solution to calculate the surface area:
1. Determine the radius:
The diameter of the hemispheres is given as 0.5 inches.
Therefore, the radius [tex]\( r \)[/tex] is:
[tex]\[
r = \frac{\text{diameter}}{2} = \frac{0.5}{2} = 0.25 \text{ inches}
\][/tex]
2. Height of the cylindrical part:
The height of the cylindrical part is equal to the diameter of the hemispheres (since it's given that diameter equals the height of the cylindrical part).
[tex]\[
h = 0.5 \text{ inches}
\][/tex]
3. Surface area of the cylindrical part:
The formula for the surface area of a cylinder (excluding the top and bottom circles) is:
[tex]\[
A_{\text{cylinder}} = 2\pi r h
\][/tex]
Substitute the values of [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[
A_{\text{cylinder}} = 2\pi \times 0.25 \times 0.5 = \frac{\pi}{2} \text{ square inches}
\][/tex]
4. Surface area of the hemispheres:
Each hemisphere has a surface area equal to half of the surface area of a sphere. The surface area of one hemisphere is:
[tex]\[
A_{\text{hemisphere}} = 2\pi r^2
\][/tex]
Since there are two hemispheres, we multiply this by 2:
[tex]\[
A_{\text{hemispheres}} = 2 \times 2\pi r^2 = 4\pi r^2
\][/tex]
Substitute the value of [tex]\( r \)[/tex]:
[tex]\[
A_{\text{hemispheres}} = 4\pi \times (0.25)^2 = 4\pi \times 0.0625 = 0.25\pi \text{ square inches}
\][/tex]
5. Total surface area of the capsule:
The total surface area [tex]\( A_{\text{capsule}} \)[/tex] is the sum of the surface area of the cylindrical part and the surface area of the hemispheres:
[tex]\[
A_{\text{capsule}} = A_{\text{cylinder}} + A_{\text{hemispheres}} = \frac{\pi}{2} + 0.25\pi = 0.5\pi + 0.25\pi = 0.75\pi \text{ square inches}
\][/tex]
6. Numerical calculation:
Convert the result to a numerical value (since [tex]\(\pi \approx 3.1416\)[/tex]):
[tex]\[
A_{\text{capsule}} = 0.75 \times 3.1416 \approx 2.3562 \text{ square inches}
\][/tex]
7. Rounding to the nearest hundredth:
The value approximately 2.3562 rounded to the nearest hundredth is 2.36.
Thus, the surface area of the capsule is:
[tex]\[
\boxed{2.36 \text{ in}^2}
\][/tex]
Therefore, the correct answer is:
C. 2.36 in [tex]$^2$[/tex]
