Question

example 2
this cylindrical coffee cup is made out of plastic. it is 5 inches tall and has a radius of 1.75 inches.
how many square inches of plastic were used to make the cup?

Explanation:

Step1: Recall the surface area formula for a cylinder (lateral + one base, since it's a cup, open at the top)

The formula for the surface area \( S \) of a cylindrical cup (open at the top) is \( S = 2\pi rh+\pi r^{2} \), where \( r \) is the radius and \( h \) is the height.
Given \( r = 1.75 \) inches and \( h = 5 \) inches.

Step2: Calculate the lateral surface area (\( 2\pi rh \))

Substitute \( r = 1.75 \) and \( h = 5 \) into \( 2\pi rh \):
\( 2\times\pi\times1.75\times5= 17.5\pi \)

Step3: Calculate the area of the base (\( \pi r^{2} \))

Substitute \( r = 1.75 \) into \( \pi r^{2} \):
\( \pi\times(1.75)^{2}=\pi\times3.0625 = 3.0625\pi \)

Step4: Sum the two areas to get the total surface area

\( S=17.5\pi + 3.0625\pi=20.5625\pi \)
Using \( \pi\approx3.14 \):
\( S\approx20.5625\times3.14 = 64.56625 \) square inches.

Answer:

Approximately \( 64.57 \) square inches (or \( 20.5625\pi \) square inches)