Question

Example 3: Solve the word problems involving angles of elevation and depression.

You are flying a kite overhead. The angle of elevation is 65°. The length of string used is 75 ft. How high is the
kite?
a.
b. Joe is standing in a bell tower 210 feet tall. He looks down at an angle of depression towards Jill who is standing
on the ground. How far is Jill from the bell tower?
Example 4: Additional Word Problems
2.
The two equal angles of an isosceles triangle are each 70°. Determine the measures of the rest of the triangle if it
has a height of 16cm.
b. A ramp leading into the public library is 25 feet long. The ramp rises a total of 2 feet. Is the ramp to code
according to ADA standards? (The angle of incline must be less than 4.76 degrees.)

Answer 1

Explanation:

3)

a. To find the height of the kite, we can use trigonometry. The sine function relates the opposite side (the height of the kite) to the hypotenuse (the length of string used) and the angle of elevation. Therefore, we can write:

sin(65°) = height/75

Solving for the height, we get:

height = 75 sin(65°) = 67.8 ft

Therefore, the kite is 67.8 feet high.

b. To find the distance between Joe and Jill, we can use trigonometry again. The tangent function relates the opposite side (the distance between Joe and Jill) to the adjacent side (the height of the bell tower) and the angle of depression. Therefore, we can write:

tan(angle of depression) = opposite/adjacent

tan(angle of depression) = Jill's height/210

Solving for the distance between Joe and Jill, we get:

distance = adjacent * tan(angle of depression)

distance = 210 * tan(angle of depression)

We need to know the angle of depression to solve for the distance, which is not given in the problem.

4)

a. In an isosceles triangle with two equal angles of 70°, the third angle must be:

180° - 70° - 70° = 40°

Since the triangle is isosceles, the height must be the perpendicular bisector of the base. Therefore, we can draw an altitude from the top vertex to the base, splitting the base into two equal segments. Let x be the length of each base segment. Then we can use trigonometry to find the height:

tan(70°) = height/x

height = x * tan(70°)

Since the height is given as 16 cm, we can solve for x:

16 = x * tan(70°)

x = 16/tan(70°)

Therefore, the length of each base segment is:

x = 16/tan(70°) = 6.12 cm

And the length of the base is twice the length of each segment:

base = 2x = 2(16/tan(70°)) = 12.25 cm

Therefore, the measures of the rest of the triangle are:

base = 12.25 cm

each equal angle = 70°

height = 16 cm

b. To determine if the ramp meets ADA standards, we need to find the angle of incline. The angle of incline is the angle between the ramp and the horizontal. We can use trigonometry to find this angle:

sin(angle of incline) = rise/run

sin(angle of incline) = 2/25

angle of incline = sin^(-1)(2/25)

Using a calculator, we get:

angle of incline ≈ 4.79°

Since the angle of incline is greater than the maximum allowable angle of 4.76°, the ramp does not meet ADA standards.