Math / Geometry

QuestionWhat is the surface area of this solid? A. 40.82 B. 47.1 C. 50.24 D. 37.68

Studdy Solution

STEP 1

1. The solid is composed of a cone on top of a cylinder.
2. The cylinder has a height of $3$ units and a radius of $1$ unit.
3. The cone has a slant height of $5$ units and a height of $4$ units.
4. The surface area includes the lateral surface area of the cone and the lateral surface area of the cylinder, but not the base of the cone since it is attached to the cylinder.

STEP 2

1. Calculate the lateral surface area of the cone.
2. Calculate the lateral surface area of the cylinder.
3. Calculate the total surface area of the solid.

STEP 3

Calculate the lateral surface area of the cone using the formula:
$\text{Lateral Surface Area of Cone} = \pi r l$
where $r$ is the radius and $l$ is the slant height. Substitute the given values:
$\text{Lateral Surface Area of Cone} = \pi \times 1 \times 5 = 5\pi$

STEP 4

Calculate the lateral surface area of the cylinder using the formula:
$\text{Lateral Surface Area of Cylinder} = 2\pi r h$
where $r$ is the radius and $h$ is the height. Substitute the given values:
$\text{Lateral Surface Area of Cylinder} = 2\pi \times 1 \times 3 = 6\pi$

STEP 5

Calculate the total surface area of the solid by adding the lateral surface areas of the cone and cylinder, and the base of the cylinder:
$\text{Total Surface Area} = 5\pi + 6\pi + \pi r^2$
Substitute the radius of the base of the cylinder:
$\text{Total Surface Area} = 5\pi + 6\pi + \pi \times 1^2 = 11\pi + \pi = 12\pi$
Convert $12\pi$ to a decimal:
$12\pi \approx 37.68$
The surface area of the solid is:
$\boxed{37.68}$

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