9. A cone has a circular base with a diameter of 18 inches and has a height of 12 inches. To the nearest square inch, what is the lateral surface area?
9. A cone has a circular base with a diameter of 18 inches and has a height of 12 inches. To the nearest square inch, what is the lateral surface area?
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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Concept explainers
Cylinders
A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.
Cones
A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.
Question
![**Problem 9: Calculating the Lateral Surface Area of a Cone**
*Question:*
A cone has a circular base with a diameter of 18 inches and a height of 12 inches. To the nearest square inch, what is the lateral surface area?
*Solution:*
The lateral surface area (LSA) of a cone can be calculated using the formula:
\[ LSA = \pi r l \]
where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone.
Given:
- The diameter of the base \( d \) = 18 inches,
- The height \( h \) = 12 inches.
First, we find the radius:
\[ r = \frac{d}{2} = \frac{18}{2} = 9 \text{ inches} \]
Next, we need to find the slant height \( l \). We can use the Pythagorean theorem for this:
\[ l = \sqrt{h^2 + r^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \text{ inches} \]
Now, calculate the lateral surface area:
\[ LSA = \pi r l = \pi \times 9 \times 15 = 135 \pi \]
To find the LSA in square inches, we can approximate π as 3.14159:
\[ LSA \approx 135 \times 3.14159 \approx 424.11 \text{ square inches} \]
Therefore, the lateral surface area of the cone, to the nearest square inch, is approximately:
\[ \boxed{424 \text{ square inches}} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a5f7bb8-38e7-4a5a-bfc7-c8f4b1290a9a%2Fdcbcebf1-1c36-42d5-8f0c-0a2fa18a9bbe%2Fkwi1oqw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 9: Calculating the Lateral Surface Area of a Cone**
*Question:*
A cone has a circular base with a diameter of 18 inches and a height of 12 inches. To the nearest square inch, what is the lateral surface area?
*Solution:*
The lateral surface area (LSA) of a cone can be calculated using the formula:
\[ LSA = \pi r l \]
where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone.
Given:
- The diameter of the base \( d \) = 18 inches,
- The height \( h \) = 12 inches.
First, we find the radius:
\[ r = \frac{d}{2} = \frac{18}{2} = 9 \text{ inches} \]
Next, we need to find the slant height \( l \). We can use the Pythagorean theorem for this:
\[ l = \sqrt{h^2 + r^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15 \text{ inches} \]
Now, calculate the lateral surface area:
\[ LSA = \pi r l = \pi \times 9 \times 15 = 135 \pi \]
To find the LSA in square inches, we can approximate π as 3.14159:
\[ LSA \approx 135 \times 3.14159 \approx 424.11 \text{ square inches} \]
Therefore, the lateral surface area of the cone, to the nearest square inch, is approximately:
\[ \boxed{424 \text{ square inches}} \]
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