Find the area of the following triangle. Round your answer to the nearest tenth. c = 210 ft, A=42.5°, B = 71.4° 20,898.3 ft2 7,721.4 f 15,442.8 ft2 21,664.4 f
Find the area of the following triangle. Round your answer to the nearest tenth. c = 210 ft, A=42.5°, B = 71.4° 20,898.3 ft2 7,721.4 f 15,442.8 ft2 21,664.4 f
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Triangle Area Calculation
In this exercise, you are required to find the area of a specific triangle. The problem provides the following information:
- Side length \( c = 210 \) feet
- Angle \( A = 42.5^\circ \)
- Angle \( B = 71.4^\circ \)
The options provided for the area of the triangle are:
- \( 20,898.3 \) square feet
- \( 7,721.4 \) square feet
- \( 15,442.8 \) square feet
- \( 21,664.4 \) square feet
#### Detailed Explanation:
To solve this, you can use trigonometric relationships and the Law of Sines to find the triangle's dimensions and then the area.
1. **Calculate Angle \( C \):**
Since the sum of all angles in a triangle is \( 180^\circ \):
\[
C = 180^\circ - A - B = 180^\circ - 42.5^\circ - 71.4^\circ = 66.1^\circ
\]
2. **Using the Law of Sines:**
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given \( c \) and angles \( A \) and \( B \), we can solve for sides \( a \) and \( b \):
\[
\frac{a}{\sin 42.5^\circ} = \frac{210}{\sin 66.1^\circ}
\]
\[
a = 210 \times \frac{\sin 42.5^\circ}{\sin 66.1^\circ}
\]
Similarly,
\[
\frac{b}{\sin 71.4^\circ} = \frac{210}{\sin 66.1^\circ}
\]
\[
b = 210 \times \frac{\sin 71.4^\circ}{\sin 66.1^\circ}
\]
3. **Calculate the Area Using the Formula:**
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2}ab \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febfa71ca-f003-4e76-b16d-d56ed3281ebe%2F1d3cf16d-747f-4318-a79f-31b29e155306%2Faebkqo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Triangle Area Calculation
In this exercise, you are required to find the area of a specific triangle. The problem provides the following information:
- Side length \( c = 210 \) feet
- Angle \( A = 42.5^\circ \)
- Angle \( B = 71.4^\circ \)
The options provided for the area of the triangle are:
- \( 20,898.3 \) square feet
- \( 7,721.4 \) square feet
- \( 15,442.8 \) square feet
- \( 21,664.4 \) square feet
#### Detailed Explanation:
To solve this, you can use trigonometric relationships and the Law of Sines to find the triangle's dimensions and then the area.
1. **Calculate Angle \( C \):**
Since the sum of all angles in a triangle is \( 180^\circ \):
\[
C = 180^\circ - A - B = 180^\circ - 42.5^\circ - 71.4^\circ = 66.1^\circ
\]
2. **Using the Law of Sines:**
The Law of Sines states:
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]
Given \( c \) and angles \( A \) and \( B \), we can solve for sides \( a \) and \( b \):
\[
\frac{a}{\sin 42.5^\circ} = \frac{210}{\sin 66.1^\circ}
\]
\[
a = 210 \times \frac{\sin 42.5^\circ}{\sin 66.1^\circ}
\]
Similarly,
\[
\frac{b}{\sin 71.4^\circ} = \frac{210}{\sin 66.1^\circ}
\]
\[
b = 210 \times \frac{\sin 71.4^\circ}{\sin 66.1^\circ}
\]
3. **Calculate the Area Using the Formula:**
The area of a triangle is given by:
\[
\text{Area} = \frac{1}{2}ab \
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