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) We can use trigonometry to find the distance of the base of the ladder from the tree. Let x be the distance we want to find.
cos(70°) = x/7
x = 7cos(70°)
x ≈ 2.30 m
Therefore, the base of the ladder is approximately 2.30 m from the tree.
b) The angle of depression is the angle between Kevin's line of sight and the horizontal. Since we know the distance to the basket and the height of the ladder, we can use trigonometry to find this angle. Let θ be the angle of depression.
tan(θ) = (height of ladder) / (distance to basket + distance from ladder base to tree)
tan(θ) = 7 / (5.4 + 2.30)
tan(θ) ≈ 0.849
θ ≈ 40°
Therefore, the angle of depression is approximately 40°.
To solve this problem, we can use trigonometric ratios, specifically the sine and cosine functions.
a) To determine the distance of the base of the ladder from the tree, we can use the sine function. The opposite side of the triangle is the distance between the basket and the base of the ladder, which is given as 5.4 m. The hypotenuse is the length of the ladder, which is 7 m. The angle between the ground and the ladder is 70°.
Using the sine function: sin(angle) = opposite / hypotenuse
sin(70°) = opposite / 7
To find the value of the opposite side (distance between the base of the ladder and the tree):
opposite = sin(70°) * 7
opposite = 0.9397 * 7
opposite ≈ 6.578 m
Therefore, the distance of the base of the ladder from the tree is approximately 6.578 m.
b) The angle of depression is the angle formed between the line of sight from Kevin's eyes to the base of the ladder and the horizontal. In this case, the angle of depression can be found using the cosine function.
Using the cosine function: cos(angle) = adjacent / hypotenuse
cos(angle) = 6.578 / 7
To find the value of the angle:
angle ≈ arccos(6.578 / 7)
angle ≈ 17.7°
Therefore, the angle of depression is approximately 18 degrees.