A landscaper sights the top of a tree at a 68 degree angle.she then moves an additional 70 feet directly away from the tree and sights the top at a 43 degree angle.how tall is the tree to the nearest tenth of a foot.
as always draw the diagram. If you review your basic trig functions, you will see that h (in feet) can be found using
h cot43° - h cot68° = 70
h = d*Tan68 = (d+70)*Tan43,
d*Tan68 = (d+70)*Tan43,
d = (d+70)*0.38,
d = 0.38d + 26.6,
d = 42.9 Ft., A = 68o.
h = d*Tan68.
To find the height of the tree, we can use the trigonometric concept of tangent.
Let's call the height of the tree "h". We can set up a right triangle with the base as the distance the landscaper moved away from the tree, which is 70 feet, and the height as "h".
In the first sighting, we can use the tangent of the angle of 68 degrees to find the height of the tree:
tan(68) = h / 70
We can rearrange this equation to solve for "h" by multiplying both sides of the equation by 70:
h = 70 * tan(68)
Using a calculator, we find that the approximate value of h is 221.905 feet (rounded to three decimal places).
Then, in the second sighting, we can use the tangent of the angle of 43 degrees to find the height of the tree:
tan(43) = h / 70
Again, rearranging the equation gives us:
h = 70 * tan(43)
Using a calculator, we find that the approximate value of h is 67.913 feet (rounded to three decimal places).
To estimate the height of the tree, we can take the average of the two height measurements:
(221.905 + 67.913) / 2 = 144.909 feet
Therefore, the height of the tree is approximately 144.9 feet (rounded to the nearest tenth of a foot).