SOLUTION: Cj needs to know the width of a river. He climbs a tree 10 meters high above the ground and finds that the angle of depression of the two opposite banks of the river is 40° and 15

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Question 1191760: Cj needs to know the width of a river. He climbs a tree 10 meters high above the ground and finds that the angle of depression of the two opposite banks of the river is 40° and 15°, respectively. How wide is the river?
Found 3 solutions by ankor@dixie-net.com, Theo, Alan3354:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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Cj needs to know the width of a river.
He climbs a tree 10 meters high above the ground and finds that the angle of depression of the two opposite banks of the river is 40° and 15°, respectively.
How wide is the river?
:
One way to do this is to solve two right triangles.
The larger triangle is formed by top of the tree and the distance from the base of the tree to the furthest bank.
The angle of depression of 15 degrees, gives an interior angle of 75 at the top of the tree. (90-15)
the distance(a)from the tree to the far river bank is the side opposite
the tree is the side adjacent
tan(75) = a%2F10%29
a = tan(75) * 10
a = 37.32 meters
:
The smaller triangle is formed by top of the tree and the distance from the base of the tree to the nearest bank.
The angle of depression of 40 degrees, gives an interior angle of 50 at the top of the tree.(90-40)
the distance (b) from the tree to the near river bank is the side opposite
the tree is the side adjacent
tan(50) = b%2F10%29
b = tan(50) * 10
b = 11.92 meters
:
The width of the river then is: 37.32 - 11.92 = 25.4 meters

Answer by Theo(13342) About Me  (Show Source):
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the angle of depression of 15 forms a right triangle with the vertical leg of length 10 and the horizontal length of x + y.
tan(15) = 10 / (x + y)
solve for (x + y) to get:
(x + y) = 10 / tan(15) = 37.3205080

the angle of depression of 40 forms a right triangle with the vertical leg of 10 and the horizontal length of x.
tan(40) = 10 / x
solve for x to get:
x = 10 / tan(40) = 11.91753593.

y is equal to x + y minus x = 25.40297215.
that is the width of the river.



let me know if you have any questions.
theo

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Only one bank is opposite.
"...angle..." should be angles.
"respectively" is superfluous.