John is painting the side of a building that is 30ft tall.
The ladder is at a 68 degree angle with the ground. while standing on the ladder the highest point John can reach is 4 ft above where the ladder touches the wall. What is the shortest ladder John can use to reach the top of the building?
I think I am suppose to use the pythagorean theorem. But he degrees is throwing me off.
Pythagorus won't help here.
Your ladder must reach up 26 ft. That means that the ladder's length x is given by
26/x = sin 68°
You're on the right track! To solve this problem, we can indeed use the Pythagorean theorem. However, we first need to consider the angle provided.
The 68-degree angle indicates the angle between the ground and the ladder. This angle is not directly related to the height of the ladder John needs to reach the top of the building. Instead, we need to find the angle formed between the ladder and the wall.
Let's call this angle θ. To determine θ, we can use the relationship between the angle of elevation and the angle of depression. In this case, the angle of elevation is 68 degrees (measured from the ground to the ladder), and the angle of depression is the angle we are trying to find (measured from the ladder to the ground).
Since the sum of the angles in a triangle is 180 degrees, we can subtract the given angle of elevation from 180 degrees to find the angle of depression.
θ = 180° - 68°
θ = 112°
Now that we have the angle θ, we can use the trigonometric ratios to find the height John needs to reach the top of the building.
The opposite side of the angle θ is the height John wants to reach (4 ft), and the adjacent side is the distance along the ground, which is the same as the length of the ladder we are looking for.
Using the tangent ratio:
tan(θ) = opposite/adjacent
tan(112°) = 4 ft/adjacent
Now, we can rearrange the equation to solve for the adjacent side (length of the ladder):
adjacent = 4 ft / tan(112°)
Adjacent ≈ 0.654 ft
Therefore, the shortest ladder John can use to reach the top of the building is approximately 0.654 feet.