Kevin is standing at the top of a ladder. The ladder is 7 m long. It is propped against a tree, and makes an
angle of 70° with the ground. To check his aim, Kevin is tossing balls into a basket located 5.4 m from the
base of the ladder, on the opposite side of the tree.
a) Determine the distance of the base of the ladder from the tree, in metres.
b) If Kevin’s eyes are even with the top of the ladder and he looks down on the basket, what is the angle of
depression? Answer to the nearest degree.
a) Using trigonometry, we can set up a ratio to solve for the distance between the base of the ladder and the tree:
tan(70°) = opposite/adjacent
tan(70°) = x/7
x = 7 * tan(70°)
x ≈ 20.33
Therefore, the distance of the base of the ladder from the tree is approximately 20.33 meters.
b) The angle of depression is the angle between the imaginary line connecting Kevin's eyes to the basket and the horizontal ground. We can use trigonometry again to solve for this angle:
tan(θ) = opposite/adjacent
tan(θ) = 5.4/20.33
θ ≈ 15.7°
Therefore, the angle of depression is approximately 15.7 degrees.
To solve this problem, we can use trigonometric functions such as sine, cosine, and tangent.
a) To determine the distance of the base of the ladder from the tree, we can use the cosine function. The cosine function relates the adjacent side to the hypotenuse. In this case, the adjacent side is the distance between the tree and the base of the ladder, and the hypotenuse is the length of the ladder.
Let's use the cosine function:
cos(70°) = adjacent side / hypotenuse
The adjacent side is the distance between the tree and the base of the ladder, which we want to find. The hypotenuse is the length of the ladder, which is 7 m.
cos(70°) = adjacent side / 7
To isolate the adjacent side, we can multiply both sides of the equation by 7:
7 * cos(70°) = adjacent side
Using a calculator, we find:
7 * 0.3420 ≈ 2.394 (rounded to three decimal places)
So, the distance of the base of the ladder from the tree is approximately 2.394 meters.
b) To find the angle of depression, we can use the tangent function. The tangent function relates the opposite side to the adjacent side.
Since Kevin is looking down on the basketball, the opposite side is the height at which Kevin's eyes are, which is equal to the height of the top of the ladder. The adjacent side is the distance between the tree and the base of the ladder, which we just found to be approximately 2.394 meters.
Let's use the tangent function:
tan(angle) = opposite side / adjacent side
We want to find the angle of depression. Let's call it θ.
tan(θ) = height of the top of the ladder / 2.394
To isolate the angle, we can take the inverse tangent of both sides of the equation:
θ = atan(height of the top of the ladder / 2.394)
Using a calculator, we find:
θ ≈ atan(height of the top of the ladder / 2.394)
θ ≈ atan(7 / 2.394)
θ ≈ atan(2.922) ≈ 70° (rounded to the nearest degree)
So, the angle of depression is approximately 70 degrees.