Question

what is the surface area of this cone? round your answer to the nearest hundredth. 64 mm 33 mm square millimeters submit

Explanation:

Step1: Recall the formula for the surface area of a cone

The surface area \( S \) of a cone is given by the sum of the lateral (or curved) surface area and the base area. The formula is \( S=\pi r l+\pi r^{2} \), where \( r \) is the radius of the base, and \( l \) is the slant height of the cone.

From the diagram, we have \( r = 33\space\text{mm} \) (radius) and \( l=64\space\text{mm} \) (slant height).

Step2: Calculate the lateral surface area (\( \pi r l \))

Substitute \( r = 33 \) and \( l=64 \) into \( \pi r l \):
\( \pi\times33\times64=33\times64\times\pi = 2112\pi \)

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

Substitute \( r = 33 \) into \( \pi r^{2} \):
\( \pi\times33^{2}=\pi\times 1089 = 1089\pi \)

Step4: Calculate the total surface area

Add the lateral surface area and the base area:
\( S=2112\pi + 1089\pi=(2112 + 1089)\pi=3201\pi \)

Now, calculate the numerical value. We know that \( \pi\approx3.14159 \)
\( S\approx3201\times3.14159 \)
First, calculate \( 3201\times3.14159 \):
\( 3201\times3.14159 = 3201\times3+3201\times0.14159=9603+3201\times0.14159 \)
\( 3201\times0.14159\approx3201\times0.14 = 448.14 \) and \( 3201\times0.00159\approx5.09 \), so \( 3201\times0.14159\approx448.14 + 5.09=453.23 \)
Then \( 9603+453.23 = 10056.23 \) (more accurately, using a calculator: \( 3201\times3.1415926535\approx3201\times3.1415926535 = 10056.28 \))

Answer:

\( 10056.28 \) (rounded to the nearest hundredth)