In a 30 ^(∘) -60 ^(∘) - 90 ^(∘) triangle, the hypotenuse h is 2 times the length of the shorter leg s, and the longer leg l is sqrt(3) times the length of the shorter leg.
We are given that the dump truck has a 15-foot bed length and asked to evaluate the height of the bed if the angle x is 30^(∘),45^(∘) or 60^(∘).
Let's start with evaluating the value of h for x=30^(∘).
x=30^(∘)
First, we will consider the case when x= 30^(∘). This means that the drawn triangle is a 30 ^(∘) -60 ^(∘) - 90 ^(∘) triangle and h is the shorter leg.
According to the 30 ^(∘) -60 ^(∘) - 90 ^(∘) Triangle Theorem, the length of the hypotenuse in the drawn triangle, which is 15, is 2 times the length of its shorter side, h.
15=2 h ⇒ h=7.5
If x=30^(∘), the height of the bed is 7.5 feet.
x=45^(∘)
Next we will consider the case when x= 45^(∘). This means that the drawn triangle is a 45 ^(∘) -45 ^(∘) - 90 ^(∘) triangle and h is the leg.
According to the 45 ^(∘) -45 ^(∘) - 90 ^(∘) Triangle Theorem, the length of the hypotenuse in the drawn triangle, which is 15, is sqrt(2) times the length of its leg, h.
15=sqrt(2) h
Let's solve the above equation for h.
If x=45^(∘), the height of the bed is approximately 10.6 feet.
x=60^(∘)
Now we will consider the case when x= 60^(∘). This means that the drawn triangle is a 30 ^(∘) -60 ^(∘) - 90 ^(∘) triangle and h is the longer leg. Let s be the length of the shorter leg.
According to the 30 ^(∘) -60 ^(∘) - 90 ^(∘) Triangle Theorem, the length of the hypotenuse in the drawn triangle, which is 15, is 2 times the length of its shorter side, s.
15=2s ⇒ s=7.5
The shorter side in this triangle is 7.5. Next let's use the fact that in the 30 ^(∘) -60 ^(∘) - 90 ^(∘) triangle the longer leg, h, is sqrt(3) times the shorter leg, 7.5.
h=sqrt(3)(7.5)≈ 13
If x=60^(∘), the height of the bed is approximately 13 feet.
