Precalculus
9th Edition
ISBN: 9780321716835
Author: Michael Sullivan
Publisher: Addison Wesley
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Question
Chapter 13, Problem 12CR
To determine
To calculate: The dimensions of a triangle and find its area where the triangle is
Expert Solution & Answer
Answer to Problem 12CR
Solution:
The sides of a triangle are a≈6.09,b=5,c=9, and angles are A=40∘,B≈31.9∘,C≈108.1∘, and area is approximately 14.46 square units.
Explanation of Solution
Given information:
The triangle:
Formula used:
Law of cosine:
a2=b2+c2−2bccosA, where a,b,c are sides of a triangle, and the angles opposite to the sides a,b,c are A,B,C respectively.
Area of triangle =12×base×height
Calculation:
Here, from the triangle, b=5,c=9, and A=40∘.
By using law of cosine,
⇒a2=52+92−2(5)(9)cos40∘
⇒a2=106−90⋅(0.7660444431)
⇒a2=106−68.94399988
⇒a2=37.05600012
⇒a=±6.087363971≈±6.09
As the length of the side is never negative, so a≈6.09.
Now, to find angle B,
By using law of cosine,
⇒b2=a2+c2−2accosB∘
⇒52=(6.09)2+92−2(6.09)(9)cosB∘
⇒25−81−37.0881=−2(6.09)(9)cosB∘
⇒−93.0881=−109.62cosB∘
⇒cosB∘=93.0881109.62
⇒cosB∘=93.0881109.62
⇒cosB∘=0.8491890166
⇒B∘=cos−1(0.8491890166)
⇒B∘=31.87642838≈31.90
⇒B∘≈31.90
Now, to find angle C,
The sum of the angles of the triangle must be equal to 180∘.
A+B+C=180
Substitute A=40∘ and B≈31.90, it gives
⇒40+31.9+C=180
⇒C=180−71.9
⇒C=108.1∘
Therefore, the sides of a triangle are a≈6.09,b=5,c=9, and angles are A=40∘,B≈31.9∘,C≈108.1∘.
Now, to find area of a triangle, The base of the triangle =9
Need to find height of the triangle, say h,
Using sine ratio to find h gives sin40∘=h5
⇒h=5sin40∘
By using the formula for area of a triangle, Area =12×9×5 sin40∘
=14.46272122≈14.46 square units.
Thus, area of a triangle is approximately 14.46 square units.
Therefore, the sides of a triangle are a≈6.09,b=5,c=9, and angles are A=40∘,B≈31.9∘,C≈108.1∘, and area is approximately 14.46 square units.
Chapter 13 Solutions
Precalculus
Ch. 13.1 - Prob. 1AYUCh. 13.1 - Prob. 2AYUCh. 13.1 - True or false The intersection of two sets is...Ch. 13.1 - Prob. 4AYUCh. 13.1 - Prob. 5AYUCh. 13.1 - Prob. 6AYUCh. 13.1 - Prob. 7AYUCh. 13.1 - Prob. 8AYUCh. 13.1 - Prob. 9AYUCh. 13.1 - Prob. 10AYU
Ch. 13.1 - If n( A )=15 , n( B )=20 , and n( AB )=10 , find...Ch. 13.1 - If n( A )=30 , n( B )=40 , and n( AB )=45 , find...Ch. 13.1 - If n( AB )=50 , n( AB )=10 , and n( B )=20 , find...Ch. 13.1 - If n( AB )=60 , n( AB )=40 , and n( A )=n( B ) ,...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - In Problems 15-22, use ihe information given in...Ch. 13.1 - A man has shirts and ties. 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