The surface area of a cone is given by the formula S = arV2 + h?, where r is the radius of the base and h is its height. Use this formula to find the number of square feet of waterproof cloth used to make the umbrella. (Assume h = 3 and r = 9.) exact answer S = ft2 approximation S = ft2
The surface area of a cone is given by the formula S = arV2 + h?, where r is the radius of the base and h is its height. Use this formula to find the number of square feet of waterproof cloth used to make the umbrella. (Assume h = 3 and r = 9.) exact answer S = ft2 approximation S = ft2
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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
![**Calculating the Surface Area of a Cone**
In this exercise, we will determine the surface area of an umbrella modeled as a cone. First, we will give the exact answer expressed as a simplified radical expression. Then, we will provide an approximation rounded to the nearest tenth.
### Formula:
The surface area \( S \) of a cone is given by the formula:
\[ S = \pi r \sqrt{r^2 + h^2} \]
where \( r \) is the radius of the base, and \( h \) is the height.
### Given:
- Radius, \( r = 9 \) feet
- Height, \( h = 3 \) feet
### Steps to Find the Surface Area:
1. **Exact Answer:**
Substitute the given values into the formula:
\[ S = \pi \times 9 \times \sqrt{9^2 + 3^2} \]
\[ S = \pi \times 9 \times \sqrt{81 + 9} \]
\[ S = \pi \times 9 \times \sqrt{90} \]
Simplify the radical if possible. Since \(\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}\), we have:
\[ S = \pi \times 9 \times 3\sqrt{10} \]
\[ S = 27\pi\sqrt{10} \]
Thus, the exact surface area, \( S \), in a simplified radical form is:
\[ S = 27\pi\sqrt{10} \ \text{ft}^2 \]
2. **Approximate Answer:**
To find an approximation, use the approximate value of \(\pi\) and the square root:
\[ S = 27 \times 3.14159 \times \sqrt{10} \]
\[ S = 27 \times 3.14159 \times 3.16228 \]
Multiply these values:
\[ S \approx 268.8 \ \text{ft}^2 \]
### Diagram:
The provided diagram shows a cone representing an opened umbrella. The height \( h \) is labeled as 3 feet, and the radius \( r \) of the circular base is labeled as 9 feet. This](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c8e8955-7b00-44a9-a272-a38b187e6aa8%2F7df4a9f8-efed-4874-a3ef-d328127116bb%2Fczx3l1_processed.png&w=3840&q=75)
Transcribed Image Text:**Calculating the Surface Area of a Cone**
In this exercise, we will determine the surface area of an umbrella modeled as a cone. First, we will give the exact answer expressed as a simplified radical expression. Then, we will provide an approximation rounded to the nearest tenth.
### Formula:
The surface area \( S \) of a cone is given by the formula:
\[ S = \pi r \sqrt{r^2 + h^2} \]
where \( r \) is the radius of the base, and \( h \) is the height.
### Given:
- Radius, \( r = 9 \) feet
- Height, \( h = 3 \) feet
### Steps to Find the Surface Area:
1. **Exact Answer:**
Substitute the given values into the formula:
\[ S = \pi \times 9 \times \sqrt{9^2 + 3^2} \]
\[ S = \pi \times 9 \times \sqrt{81 + 9} \]
\[ S = \pi \times 9 \times \sqrt{90} \]
Simplify the radical if possible. Since \(\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}\), we have:
\[ S = \pi \times 9 \times 3\sqrt{10} \]
\[ S = 27\pi\sqrt{10} \]
Thus, the exact surface area, \( S \), in a simplified radical form is:
\[ S = 27\pi\sqrt{10} \ \text{ft}^2 \]
2. **Approximate Answer:**
To find an approximation, use the approximate value of \(\pi\) and the square root:
\[ S = 27 \times 3.14159 \times \sqrt{10} \]
\[ S = 27 \times 3.14159 \times 3.16228 \]
Multiply these values:
\[ S \approx 268.8 \ \text{ft}^2 \]
### Diagram:
The provided diagram shows a cone representing an opened umbrella. The height \( h \) is labeled as 3 feet, and the radius \( r \) of the circular base is labeled as 9 feet. This
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