Question

Please help me with this problem?

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(t)=42+sin((pit)/12)`.
The height of Peter's cabin is given by `h_P(t)=42-cos((pit)/12)`.
Time is measured in minutes, height in meters.
At time interval `[0,24]`, for which `t` do we have `h_P(t)>h_J(t)`?

Answer 1

`9 < t < 21`

Explanation:

Given:

`h_J(t) = 42+sin((pit)/12)`

`h_P(t) = 42-cos((pit)/12)`

We want to know when:

`h_P(t) > h_J(t)`

That is:

`42-cos((pit)/12) > 42+sin((pit)/12)`

Subtract `42-cos((pit)/12)` from both sides to get:

`0 > sin((pit)/12) + cos((pit)/12)`

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. `pi/4`, `(3pi)/4`, `(5pi)/4`, `(7pi)/4`.

Of these, the quadrants in which `sin` and `cos` are of opposite signs are Q2 and Q4, with midpoints `(3pi)/4` and `(7pi)/4`

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

Note that `sin(0)+cos(0) = 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(t) > h_J(t)` is:

`(3pi)/4 < (pit)/12 < (7pi)/4`

Multiply through by `12/pi` to get:

`9 < t < 21`