Problem 2:Two bumper cars, one blue(left) and one orange(right), at an amusement park ridecollide as shown below.7 ft118°5.5 fta. How far apart d were the two cars before they collided?b. Before the collision, a third red car was 10 feet from the blue car and 13feet from the orange car. Describe the angles formed at all three cars beforethe collision.

Part A. In the given problem a triangle is formed. Since we are required to determine one of its sides given the opposite angle and the other two sides we can use the cosine law:

[tex]c^2=a^2+b^2-2ab\cos\theta[/tex]

Where:

[tex]\begin{gathered} c,a,b\text{ = sides} \\ \theta=\text{ opposite angle} \end{gathered}[/tex]

Now, we plug in the values:

[tex]d^2=7^2+5.5^2-2(7)(5.5)\cos118[/tex]

Now, we solve the operations:

[tex]d^2=115.4[/tex]

Now, we take the square root to both sides:

[tex]\begin{gathered} d=\sqrt{115.4} \\ d=10.7 \end{gathered}[/tex]

Therefore, the distance between the cars is 10.7 feet.

Part B. We are given the following situation:

We will apply the coside law using the angle "x" as the opposite angle:

[tex]10.7^2=10^2+13^2-2(10)(13)\cos x[/tex]

Now, we solve the operations:

[tex]114.49=100+169-260\cos x[/tex]

Now, we solve the addition:

[tex]114.49=269-260\cos x[/tex]

Now, we subtract 269 from both sides:

[tex]\begin{gathered} 114.49-269=-260\cos x \\ -145.51=-260\cos x \end{gathered}[/tex]

Now, we divide both sides by -260:

[tex]\frac{-145.51}{-260}=\cos x[/tex]

Now, we take the inverse function of the cosine:

[tex]\cos^{-1}(\frac{-145.51}{-260})=x[/tex]

Solving the operations:

[tex]55.97=x[/tex]

Therefore, angle "x" is 55.97 degrees.

Now, we use angle "y" as the opposite angle. Applying the cosine law we get:

[tex]13^2=10^2+10.7^2-2(10)(10.7)\cos y[/tex]

Solving the operations:

[tex]169=214.5-214\cos y[/tex]

Now, we subtract 214.5 from both sides:

[tex]\begin{gathered} 169-214.5=-214\cos y \\ -45.5=-214\cos y \end{gathered}[/tex]

Now, we divide both sides by 214

[tex]\frac{-45.5}{-214}=\cos y[/tex]

Now, we take the inverse function of the cosine:

[tex]\begin{gathered} \cos^{-1}(-\frac{45.5}{-214})=y \\ \\ 77.75=y \end{gathered}[/tex]

Therefore, angle "y" is 77.72 °

To determine angle "z" we will use the fact that the sum of the interior angles of a triangle always adds up to 180:

[tex]\begin{gathered} x+y+z=180 \\ \end{gathered}[/tex]

Plugging in the values:

[tex]55.97+77.72+z=180[/tex]

Solving the operations:

[tex]133.69+z=180[/tex]

Now, we subtract 133.69 from both sides:

[tex]\begin{gathered} z=180-133.69 \\ z=46.31 \end{gathered}[/tex]

Therefore, angle "z" is 46.31 degrees.