Question

select the correct answer.
the shape of a capsule consists of a cylinder with identical hemispheres on each end. the diameter of the hemispheres is 0.5 inches.
what is the surface area of the capsule? round your answer to the nearest hundredth.
a. 3.14 in²
b. 6.28 in²
c. 2.36 in²
d. 3.93 in²

Explanation:

Step1: Find the radius of the hemispheres

The diameter \(d = 0.5\) inches. The radius \(r=\frac{d}{2}=\frac{0.5}{2}=0.25\) inches.

Step2: Calculate the surface area of the two hemispheres (which is equivalent to the surface area of a full - sphere)

The formula for the surface area of a sphere is \(A_{sphere}=4\pi r^{2}\). Substitute \(r = 0.25\) into the formula: \(A_{sphere}=4\pi(0.25)^{2}=4\pi\times0.0625=\pi\approx3.14\) square inches.

Step3: Calculate the lateral surface area of the cylinder

The formula for the lateral surface area of a cylinder is \(A_{lateral}=2\pi r h\). The height \(h\) of the cylinder: The total length of the capsule is \(1\) inch. The combined length of the two hemispheres (in terms of their diameter) is \(0.5\) inches. So the height of the cylinder \(h=1 - 0.5=0.5\) inches. Substitute \(r = 0.25\) and \(h = 0.5\) into the formula: \(A_{lateral}=2\pi\times0.25\times0.5=\pi\times0.25\approx0.79\) square inches.

Step4: Calculate the total surface area of the capsule

The total surface area \(A = A_{sphere}+A_{lateral}\). So \(A=\pi+0.25\pi = 1.25\pi\approx3.93\) square inches.

Answer:

D. \(3.93\ in^{2}\)