a First determine which sides of the right triangles are being discussed.
B
b First determine which sides of the right triangles are being discussed.
A
a About 2.7 feet
B
b About 1.3 feet
Practice makes perfect
a We are given two bicycle ramps as shown.
We want to find how much taller the second ramp is than the first. From the graphs we can see that the mentioned lengths correspond to the opposite sides of the given angles. Let's name the desired sides h_1 and h_2.
We will write equations that include the ratio of the opposite sides and the adjacent sides of the given angles. Therefore, we will use the tangent of an acute angle θ, which is the ratio between the lengths of the opposite side and the adjacent side.
tan θ = Opposite/Adjacent [1em]
⇓
[0.5em]
tan 20^(∘) = h_1 8 and tan 35^(∘) = h_2 8
Now we will solve the equations one by one. Let's begin by solving h_1.
The height of the second ramp is approximately 5.6 feet. We want to find how much taller h_2 is than h_1.
h_2- h_1&= 5.6- 2.9
&=2.7 feet
Therefore, the second ramp is about 2.7 feet taller than the first one.
b Now we want to find how much longer the second ramp is than the first.
From the graphs we can see that the desired lengths correspond to the hypotenuses. Let's name the hypotenuses x_1 and x_2.
We will write equations that include the ratio of the adjacent sides and the hypotenuses of the given angles. Therefore, we will use the cosine of an acute angle θ, which is the ratio between the lengths of the adjacent side and the hypotenuse.
cos θ = Adjacent/Hypotenuse [1em]
⇓
[0.5em]
cos 20^(∘) = 8 x_1 and cos 35^(∘) = 8 x_2
Now we will solve the equations one by one. Let's begin by solving x_1.
The length of the second ramp is approximately 9.8 feet. We want to find how much longer x_2 is than x_1.
x_2- x_1&= 9.8- 8.5
&=1.3 feet
Therefore, the second ramp is about 1.3 feet longer than the first one.
