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Chapter 3: Problem 32

A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries?

Short Answer

Expert verified
The angles between the frontage and the other two boundaries are calculated with the law of cosines and expressed in degrees. For precise values, it is advised to use a calculator or mathematical software, especially for inverse cosine calculations.

Step by step solution

01

Determine the sides and their opposite angles

Label the sides of the triangle as a = 115m (frontage), b = 76m and c = 92m. The angles opposite these sides are denoted as A, B, C correspondingly.
02

Use the law of cosines for first angle

The law of cosines is given by: \( a^2 = b^2 + c^2 - 2bc \cdot cosA \). Rearrange the equation to solve for angle A: \( cosA = (b^2 + c^2 - a^2) / (2bc) \). Substitute the values: \( cosA = (76^2 + 92^2 - 115^2) / (2*76*92) \). Compute the value of \( cosA \) and use the inverse cosine function to find the angle in degrees.
03

Use the law of cosines for second angle

Next, use the same approach to find angle B: \( cosB = (a^2 + c^2 - b^2) / (2ac) \). Substitute the values: \( cosB = (115^2 + 92^2 - 76^2) / (2*115*92) \). Compute the value of \( cosB \) and use the inverse cosine function to find the angle in degrees.
04

Compute the last angle

By knowing two angles in a triangle, the third can be easily found since the sum of the angles in a triangle equals 180 degrees. Compute angle C as \( C = 180 - A - B \).

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