Find the area of the following triangle. Round your answer to the nearest tenth. c = 210 ft, A = 42.5°, B = 71.4° 21,664.4 ff O 15,442.8 ft2 20,898.3 ft2 O 7,721.4 ff
Find the area of the following triangle. Round your answer to the nearest tenth. c = 210 ft, A = 42.5°, B = 71.4° 21,664.4 ff O 15,442.8 ft2 20,898.3 ft2 O 7,721.4 ff
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
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![### Calculating the Area of a Triangle
In this problem, we'll learn how to calculate the area of a triangle when given specific measurements.
**Given**:
- Side length \( c = 210 \) feet
- Angle \( A = 42.5^\circ \)
- Angle \( B = 71.4^\circ \)
**Problem**:
Find the area of the following triangle to the nearest tenth of a square foot.
**Options**:
1. \( 21,664.4 \, \text{ft}^2 \)
2. \( 15,442.8 \, \text{ft}^2 \)
3. \( 20,898.3 \, \text{ft}^2 \)
4. \( 7,721.4 \, \text{ft}^2 \)
**Approach**:
The area of a triangle can be calculated using various methods. One effective method for cases where two angles and one side are known is using the formula:
\[ \text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \]
However, before we can use this formula, we need to determine the length of the other sides using the given angles and side. We can use the Law of Sines:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
In this setup, since angle \( C \) can be found as:
\[ C = 180^\circ - A - B \]
Let's plug in the values and solve the problem using these trigonometric relationships. Select the appropriate method based on your learning material's presentation and guidelines.
**Options provided**, one will be nearest to the computed area using the appropriate triangle area formulas and laws.
This interactive problem allows you to apply principles of trigonometry and geometry to solve real-world problems effectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc7859e1-a84c-4143-bc4d-4794151a052e%2F40d98531-9017-4ce7-b99c-cd5e65515c05%2Fi66whp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Calculating the Area of a Triangle
In this problem, we'll learn how to calculate the area of a triangle when given specific measurements.
**Given**:
- Side length \( c = 210 \) feet
- Angle \( A = 42.5^\circ \)
- Angle \( B = 71.4^\circ \)
**Problem**:
Find the area of the following triangle to the nearest tenth of a square foot.
**Options**:
1. \( 21,664.4 \, \text{ft}^2 \)
2. \( 15,442.8 \, \text{ft}^2 \)
3. \( 20,898.3 \, \text{ft}^2 \)
4. \( 7,721.4 \, \text{ft}^2 \)
**Approach**:
The area of a triangle can be calculated using various methods. One effective method for cases where two angles and one side are known is using the formula:
\[ \text{Area} = \frac{1}{2} \cdot a \cdot b \cdot \sin(C) \]
However, before we can use this formula, we need to determine the length of the other sides using the given angles and side. We can use the Law of Sines:
\[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
In this setup, since angle \( C \) can be found as:
\[ C = 180^\circ - A - B \]
Let's plug in the values and solve the problem using these trigonometric relationships. Select the appropriate method based on your learning material's presentation and guidelines.
**Options provided**, one will be nearest to the computed area using the appropriate triangle area formulas and laws.
This interactive problem allows you to apply principles of trigonometry and geometry to solve real-world problems effectively.
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