A 35 -ft ladder leans against a building so that the angle between the ground and the ladder is 82 ∘ . How high does the ladder reach on the building? Round to the hundredths.

Answer 1

We can find a solution using the following three concepts:

1) Sum of the interior angles in a triangle can not exceed 180°

2) Right triangles have one interior angle of 90°

3) Law of Sines a/sinA = b/sinB = c/sinC

Part 1)

The problem explicitly says one interior angle is 82° (between the ground and base of the ladder). Using concept 1 & 2 will allow us to find the interior angle between the top of the ladder and the building. Given this is a right triangle, we know the angle between the ground and building is 90°.

82° + 90° = 172°

Part 2)

We then subtract the sum of our know interior angles from 180° to find the unknown interior angle.

180° - 172° = 8°

Part 3)

We should always verify our angles equal 180°.

82° + 90° + 8° = 180°

Part 4) a/sinA = b/sinB = c/sinC

We are now ready to use concept 3 (Law of Sines) to solve for the height of the building. We will imgine a simple ABC triangle.

With angles:

Angle A = (8°) Top of building and ladder

Angle B = (82°) Base of ladder and ground

Angle C = (90°) Ground and base of building

And lengths (sides of the triangle):

length a = unk (between angles B & C) aka the area of ground under the ladder.

length b= unk (between angles C & A) aka the height of the building

length c = 35ft (between angles A & B) aka the length of the ladder

The problem gave us one length, the ladder (35 ft), which is our length c. Now that we have all the information gathered we can start solving the problem using the Law of Sines formula.

We want to know length b and we have information for, Angle C (90°), length c (35ft), & Angle B (82°). so we set the equation up such as: c/sinC = b/sinB

Step 1: Set up

35/sin(90) = b/sin(82)

Step 2: Multiple sin(82) to both sides

sin(82) * 35/sin(90) = b/sin(82) * sin(82)

Step 3: Plug into calculator

sin(82)*35/sin(90) = b

Step 4: Solution

b= 34.65938241 feet

You could have skipped Part 1 - Part 3 but it's a good habit to always check the answer, which we can do with the information obtained in these parts. We can use the Law of Sines formula to check our answer.

This time using a/sinA = c/sinC

4.8710/sin(8) = c/sin(90)

sin(90)*4.8710/sin(8) = c/sin(90)*sin(90)

sin(90)*4.8710/sin(8) = c

c=34.999999