## John and Peter take a ride in a ferris wheel. John enters first, Peter after 6 minutes. Measuring from the time that Peter enters, the height of John's cabin as a function of time is given by: ${h}_{J} \left(t\right) = 42 + \sin \left(\frac{\pi t}{12}\right)$. The height of Peter's cabin is given by ${h}_{P} \left(t\right) = 42 - \cos \left(\frac{\pi t}{12}\right)$. Time is measured in minutes, height in meters. At time interval $\left[0 , 24\right]$, for which $t$ do we have ${h}_{P} \left(t\right) > {h}_{J} \left(t\right)$?

Jul 25, 2017

$9 < t < 21$

#### Explanation:

Given:

${h}_{J} \left(t\right) = 42 + \sin \left(\frac{\pi t}{12}\right)$

${h}_{P} \left(t\right) = 42 - \cos \left(\frac{\pi t}{12}\right)$

We want to know when:

${h}_{P} \left(t\right) > {h}_{J} \left(t\right)$

That is:

$42 - \cos \left(\frac{\pi t}{12}\right) > 42 + \sin \left(\frac{\pi t}{12}\right)$

Subtract $42 - \cos \left(\frac{\pi t}{12}\right)$ from both sides to get:

$0 > \sin \left(\frac{\pi t}{12}\right) + \cos \left(\frac{\pi t}{12}\right)$

In order for the right hand side to be zero, we would require $\sin$ and $\cos$ to be the same size but of opposite signs.

The angles at which $\sin \theta$ and $\cos \theta$ are of equal size are the midpoints of the four quadrants, i.e. $\frac{\pi}{4}$, $\frac{3 \pi}{4}$, $\frac{5 \pi}{4}$, $\frac{7 \pi}{4}$.

Of these, the quadrants in which $\sin$ and $\cos$ are of opposite signs are Q2 and Q4, with midpoints $\frac{3 \pi}{4}$ and $\frac{7 \pi}{4}$

Hence the sign of $\sin \theta + \cos \theta$ changes at these two points in $\left[0 , 2 \pi\right]$.

Note that $\sin \left(0\right) + \cos \left(0\right) = 0 + 1 = 1 > 0$. So the inequality is not satisfied at $t = 0$. This is what we would expect, since the entrance to the Ferris wheel is at the lowest point.

Hence the interval for which ${h}_{P} \left(t\right) > {h}_{J} \left(t\right)$ is:

$\frac{3 \pi}{4} < \frac{\pi t}{12} < \frac{7 \pi}{4}$

Multiply through by $\frac{12}{\pi}$ to get:

$9 < t < 21$