# A surveyor measure the distance across a straight river by the following methods:starting across from a tree on the opposite bank. she walks 278malong the river bank to establish a baseline. Then...

A surveyor measure the distance across a straight river by the following methods:starting across from a tree on the opposite bank. she walks 278m

along the river bank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is 44.7 degrees. How wide is the river? Answer in units of m.

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The easiest way to find the width of the river is to use the tangent function of a right triangle.  The base of your triangle is the 278 m that was measured along the river bank. The vertical side opposite the measured angle is the distance you are trying to find.

The tangent of an angle equals the opposite side divided by the adjacent side of the triangle.  Remember SOH CAH TOA?  Tangent = opposite/adjacent.

Using this relationship, the tangent of 44.7 degrees = x/278 m.

x = 278 tan 44.7 = 275.103 m.  Just be sure you are in degrees mode on your calculator, not radians!

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In this problem three points - that is the position of the tree (A), the original position of the surveyor (B), and the position of surveyor after walking 278 m (C) forms a right angle triangle with angle ABC being a right angle.

We know the length BC (278 m) and angle BCA (44.7 degrees).

We have to find distance betewwn tree and original position of surveyor. This is equal to AB

tan(BCA) = tan(44.7) = (AB)/(BC)

Substituting value of tan 44.7 as 0.9896 and given value of BC:

0.9896 = (AB)/278

Therefore:

AB = 0.9896*278 = 275.1088 m