Question 112229: I need help solving this question:
From a boat on the water, the angle of elevation to the top of a cliff is 31 degrees. Another point situated 300 meters closer to the cliff has an angle of elevation of 33 degrees. What is the height of the cliff?
I was trying to draw a diagram but I'm not sure if it was right, can you please show your work so I know what I'm doing wrong?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let's make a sketch as follows:
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On a piece of paper draw a horizontal line about 3 inches long. Label the left end of this
line as point A. Label the right end of this line as point B. Next through point B draw a
perpendicular to line AB vertically upward for a couple of inches. Label the top of this
line as point C. Draw a line connecting point A with point C, which forms a right triangle
ABC with line AC being the hypotenuse. Label angle BAC as 31 degrees.
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Next, on line AB make a point about an inch to the right of point A. Label this point as
point D. Next draw a line connecting point D with point C. Notice that triangle BDC is another
right triangle. Its hypotenuse is line DC. Label angle BDC as 33 degrees
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Label the length AD as 300 meters.
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Now understand that point A is the first position of the boat. Point C is the top of the
cliff. Line CB represents the height of the cliff. Point D is the second position
of the boat ... 300 meters closer to the base of the cliff. The two angles (31 degrees
and 33 degrees) show what happens as you get closer to the base of the cliff. You have to
have a greater angle for your line of sight to be on the top of the cliff.
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Now let's use the tangent function. The tangent ratio is the side opposite the angle divided
by the side adjacent to the angle. The side opposite the 31 degree angle is side CB.
And the side adjacent to the 31 degree angle is side AB. So we can write:
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tan(31) = CB/AB
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Using a calculator you can find that tan(31) = 0.600860619
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So the equation becomes:
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0.600860619 = CB/AB
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and when you multiply both sides by AB you get:
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0.600860619AB = CB <=== call this equation #1
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Now lets look at triangle BDC and angle BDC (33 degrees). In this triangle, the side opposite
angle BDC is side CB. The side adjacent to angle BDC is side BD. But side BD is equal
to side AB minus 300 meters (length AD). So for the tangent of angle BDC (33 degrees) we
get:
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tan(33) = CB/(AB - 300)
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Using a calculator we find that tan(33) = 0.649407593
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Substituting this we get:
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0.649407593 = CB/(AB - 300)
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Multiplying both sides by (AB - 300) results in:
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0.649407593(AB - 300) = CB
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And multiplying out the left side results in:
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0.649407593AB - 194.822278 = CB <==== call this equation #2
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Notice now that equations #1 and #2 both have CB on the right side. This means that the
left sides of these equations both equal CB and therefore, the left sides of the two equations
must be equal. So set them equal to get:
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0.600860619AB = 0.649407593AB - 194.822278
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Subtract 0.649407593AB from both sides and you get rid of the 0.649407593AB on the right
side. The equation then becomes:
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-0.048546974AB = -194.822278
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Divide both sides by -0.048546974 and you get:
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AB = -194.822278/-0.048546974 = 4013.067384 meters
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This is the distance from the first position of the boat to the base of the cliff. Equation
#1 applies to that position and we can substitute 4013.067384 into that equation for AB to get:
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.600860619(4013.067384) = CB
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Multiplying out the right side results in 2411.294153 meters = CB
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Or the height of the cliff is about 2411 meters ... pretty tall!!
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Hope this helps you to understand the problem and one way you can do it.
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