Use the Law of Cosines to find the distance between the two bumper cars before the collision.
B
b
Illustrate the situation on a diagram. Use the Law of Cosines to find the first angle.
A
a
10.7 feet
B
b
54^(∘), 49^(∘), and 77^(∘)
Practice makes perfect
a
We are given a diagram showing how two bumper cars at an amusement park collide.
We want to find how far apart the cars were before they collided. Therefore, we need to find the value of d. Let's use the Law of Cosines to do so.
For our triangle, the known side lengths are 5.5 and 7, the included angle measure is 118^(∘), and the missing side length is d. Let's substitute these values into the Law of Cosines and find the value of d.
Note that since a negative side length does not make sense, we only need to consider positive solutions. The two bumper cars were about 10.7 feet apart before they collided.
b
We are told that before the collision a third car was 10 feet from the blue car and 13 feet from the orange car. Let's illustrate this on our diagram. We will mark the third car with red and also name each car with a letter.
We want to find the measures of the angles formed by the three cars before the collision. We will find these angle measures one at a time.
Finding m∠B
Let's start with finding the measure of the angle ∠B. To do so, we can use the Law of Cosines.
b^2= a^2+ c^2-2 a ccos B
Let's substitute a= 13, b= 10.7, and c= 10 to isolate cos B.
Finally, to find m ∠A we can use the Triangle Angle Sum Theorem. This tells us that the measures of the angles in a triangle add up to 180.
m ∠A+ 54+ 49 = 180 ⇔ m ∠A ≈ 77
Finally, we have that the angles between the three cars before the collision were 54^(∘), 49^(∘) and 77^(∘).
