SOLUTION: a surveyor wishes to find the distance between two inaccessible points A&B on opposite side of the lake while standing at point C she finds that AC=259m ,BC=423inches and angle ABC

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Question 392634: a surveyor wishes to find the distance between two inaccessible points A&B on opposite side of the lake while standing at point C she finds that AC=259m ,BC=423inches and angle ABC measures 132 40' find the distance AB?
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
The Law of Cosines has four variables: the lengths of the 3 sides of a triangle and the measure of one angle.

Your problem gives you 2 sides and the measure of one angle. So you know the values 3 of the 4 variables in the Law of Cosines. Anytime you know all but 1 variable in an equation you should be able to solve for the value of that last unknown. So we will use the Law of Cosines to solve this problem:

To convert BC into meters, we divide by 39.37:
BC = 423 inches = 432/39.37 meters = 10.7442214884429769 meters.
Also 132 degrees and 40 minutes = 132.67 degrees since 40 minutes is 2/3 of a degree and 2/3 as a decimal is approximately 0.67.
Inserting these values into our Law of Cosines equation we get:

To avoid confusion I am going to replace (AB) with an "x":

Simplifying we get:

This is a quadratic equation because of the . SO we want one side to be zero. Subtracting 67081 from each side we get:

Now we can use the Quadratic Formula:

which simplifies as follows:

In long form this is:
or
We can see that the second equation will give us a negative value. But the side of a triangle cannot be negative. So we will reject the second solution. The only solution we can use is:

Using a calculator to find the square root we get:

x = 251.4661081380912442
So AB is approximately 251.5 meters long.