Question

the total surface area is □ in.² (round to the nearest integer as needed.) find the total surface area and volume of the frustum of a cone.

Explanation:

Step1: Identify the formula for the total surface area of a frustum of a cone

The total surface area \( A \) of a frustum of a cone is given by the formula:

$$ A = \pi (R + r) l + \pi R^2 + \pi r^2 $$

where \( R \) is the radius of the larger base, \( r \) is the radius of the smaller base, and \( l \) is the slant height of the frustum.

From the diagram:

• The diameter of the larger base is \( 15'' \), so \( R=\frac{15}{2}=7.5'' \)
• The diameter of the smaller base is \( 10'' \), so \( r=\frac{10}{2}=5'' \)
• The slant height \( l = 13.2'' \) (given)

Step2: Calculate the lateral (curved) surface area

The lateral surface area \( L \) is \( \pi (R + r) l \)

$$ L=\pi(7.5 + 5)\times13.2=\pi\times12.5\times13.2 $$

$$ L = 165\pi \approx 165\times3.1416\approx518.36 $$

Step3: Calculate the area of the larger base

The area of the larger base \( A_R=\pi R^2 \)

$$ A_R=\pi\times(7.5)^2=\pi\times56.25\approx176.71 $$

Step4: Calculate the area of the smaller base

The area of the smaller base \( A_r=\pi r^2 \)

$$ A_r=\pi\times(5)^2=\pi\times25\approx78.54 $$

Step5: Calculate the total surface area

The total surface area \( A = L+A_R + A_r \)

$$ A\approx518.36 + 176.71+78.54 $$

$$ A\approx773.61 $$

Answer:

\( 774 \) (rounded to the nearest integer)