Question

question 10 of 21
what is the surface area of this solid?
a. 226.08
b. 257.36
c. 163.28
d. 150.72

Explanation:

Step1: Identify the solid components

The solid is a combination of a cylinder and a cone. Let's assume the radius \( r = 2 \) (from the diagram, the green part's radius, and the cone's base radius), height of cylinder \( h_{cyl} = 6 \), slant height of cone \( l = 10 \) (given in the diagram for the cone).

Step2: Surface area of cylinder (lateral + one base, since the top is attached to the cone)

Lateral surface area of cylinder: \( 2\pi rh_{cyl} \)
Area of one base of cylinder: \( \pi r^2 \)
So cylinder surface area: \( 2\pi rh_{cyl} + \pi r^2 \)
Substitute \( r = 2 \), \( h_{cyl} = 6 \):
\( 2\pi(2)(6) + \pi(2)^2 = 24\pi + 4\pi = 28\pi \)

Step3: Lateral surface area of cone

Lateral surface area of cone: \( \pi rl \)
Substitute \( r = 2 \), \( l = 10 \):
\( \pi(2)(10) = 20\pi \)

Step4: Total surface area

Add cylinder and cone lateral areas (and cylinder's base):
Total \( = 28\pi + 20\pi = 48\pi \)
Calculate \( 48\pi \approx 48\times3.1416 \approx 150.72 \) (Wait, but let's check again. Wait, maybe the cylinder's height is 6, radius 2, cone slant height 10, radius 2. Wait, maybe I made a mistake. Wait, let's re - evaluate.

Wait, maybe the cylinder has height 6, radius 2. The cone has radius 2, slant height 10.

Surface area of the composite solid:

• Cylinder: lateral surface area \( 2\pi rh = 2\times\pi\times2\times6 = 24\pi \), and the area of the bottom base \( \pi r^2=\pi\times2^2 = 4\pi \). The top base of the cylinder is internal (attached to the cone), so we don't include it.

• Cone: lateral surface area \( \pi rl=\pi\times2\times10 = 20\pi \). The base of the cone is attached to the cylinder, so we don't include it.

Total surface area: \( 24\pi + 4\pi+ 20\pi=48\pi\approx48\times3.14 = 150.72 \)

Answer:

D. 150.72