Chapter 12.3, Problem 5GP

To determine

To find: Total height of the ride at the peak of the arc.

Answer to Problem 5GP

Total height of the ride at its peak $=106.6$ feet.

Explanation of Solution

Given information:

Swing arms of the ride at the right are $72$ feet long.

Height of the axis from which the arms swing is $88$ feet.

Angle is $-{195}^0$

Concept:

If $\theta$ is the non-quadrantal angle in standard position, its reference angle ${\theta }^{\text{'}}$ is the acute angle formed by the terminal side of $\theta$ and the x-axis.

The ruled for finding the measures of reference angles for $0^0<\theta <{360}^0$ or $0^0<\theta <2\pi$ are shown

For Quadrant − I

  ${\theta }^{\text{'}}=\theta$

For Quadrant − II

  ${\theta }^{\text{'}}={180}^0-\theta$ or ${\theta }^{\text{'}}=\pi -\theta$

For Quadrant − III

  ${\theta }^{\text{'}}=\theta -{180}^0$ or ${\theta }^{\text{'}}=\theta -\pi$

For Quadrant − IV

  ${\theta }^{\text{'}}={360}^0-\theta$ or ${\theta }^{\text{'}}=2\pi -\theta$

Given

Angle is $-{195}^0$

Co-terminal angle is

  $\theta =-{195}^0+{360}^0={165}^0$

Terminal side $\theta ={165}^0$ lies in Quadrant-II

Therefore,

Reference angle is

  $\begin{array}{l}{\theta }^{\text{'}}={180}^0-{165}^0\\ {\theta }^{\text{'}}={15}^0\end{array}$

We know that

  $\sin \theta =\frac{y}{r}$

On substituting values

We get

  $\begin{array}{l}\Rightarrow \sin \left({15}^0\right)=\frac{y}{72}\\ \Rightarrow y=72\times \sin \left({15}^0\right)\\ \Rightarrow y=72\times \left(0.2589\right)\\ \Rightarrow y=18.635\end{array}$

Since $y$ is approximately $18.6$ feet, the total height of the ride at its peak is $18.6+88$ or about $106.6$ .

Therefore,

Total height of the ride at its peak $=106.6$ feet.