Answer: 30 feet
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Step-by-step explanation:
It might help to erase any unneeded stuff from the drawing. Refer to the diagram below to see what I mean. We have two triangles and we need to find the lengths of BD and DC, which are x and y respectively.
To find each of them, we'll use the tangent ratio since it connects the opposite and adjacent sides together.
Focus on triangle ADB up top
tan(angle) = opposite/adjacent
tan(A) = BD/AD
tan(42) = x/20
20*tan(42) = x
x = 20*tan(42)
x = 18.0080808859568 which is approximate.
x = 18.008
So the length of segment BD is roughly 18.008 feet.
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We'll use this same idea to find y. Focus on triangle ADC.
tan(angle) = opposite/adjacent
tan(A) = DC/AD
tan(31) = y/20
20*tan(31) = y
y = 20*tan(31)
y = 12.0172123805512 which is also approximate
y = 12.017
Segment DC is approximately 12.017 feet
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At the end of the each of the last two sections we found that
segment BD = 18.008 feet
segment DC = 12.017 feet
add those two segments together to get the length of segment BC, which is the height of the tree
BC = BD+DC
BC = 18.008 + 12.017
BC = 30.025
BC = 30 feet, which is the final answer
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If you wanted to calculate everything in nearly one step, then you could say
20*tan(42)+20*tan(31) = 30.02529 = 30
This one step calculation is simply a quick summary of what the last two sections are talking about in more step by step detail.