To determine the surface area of the capsule, we need to consider the shape as consisting of a cylinder and two hemispheres at each end. Given that the diameter of the hemispheres is 0.5 inches, we can calculate the surface area step-by-step.
1. Calculate the radius of the hemispheres:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{0.5 \text{ inches}}{2} = 0.25 \text{ inches}
\][/tex]
2. Surface area of the cylinder (excluding the bases):
For this particular problem, we assume the height of the cylinder part is negligible or 0 inches, hence the surface area of the cylinder part can be considered zero.
[tex]\[
\text{Surface area of the cylinder} = 2 \pi (\text{Radius}) (\text{Height}) = 0
\][/tex]
3. Surface area of the two hemispheres:
A full sphere’s surface area is:
[tex]\[
4 \pi (\text{Radius})^2
\][/tex]
Since we have two hemispheres, together they make up one full sphere. Thus, the surface area of both hemispheres is:
[tex]\[
\text{Surface area of the hemispheres} = 4 \pi (0.25)^2
\][/tex]
Calculating this surface area:
[tex]\[
\text{Surface area of the hemispheres} = 4 \pi (0.0625)
\][/tex]
[tex]\[
\text{Surface area of the hemispheres} = 4 \cdot 3.14159 \cdot 0.0625 = 0.79 \text{ square inches}
\][/tex]
4. Total surface area of the capsule:
Summing the surface area of the cylinder and the hemispheres:
[tex]\[
\text{Total surface area} = \text{Surface area of the cylinder} + \text{Surface area of the hemispheres}
\][/tex]
As the cylinder's area is zero:
[tex]\[
\text{Total surface area} = 0 + 0.79 = 0.79 \text{ square inches}
\][/tex]
So, the surface area of the capsule, rounded to the nearest hundredth, is:
[tex]\[
\boxed{0.79 \text{ in}^2}
\][/tex]
Given the options provided:
- A. 6.28 in [tex]$^2$[/tex]
- E. 3.93 in [tex]$^2$[/tex]
- C. [tex]$3.14 in ^2$[/tex]
- D. [tex]$2.36 in ^2$[/tex]
None of the provided options match the correct calculated value. The correct answer of [tex]\(0.79 \text{ in}^2\)[/tex] is not among the choices provided in the question.
