Question

I need help with a pre-calc problem I really don't understand how to solve it

(the answer is: 101.496936 feet above the ground.)
And please explain how you got the answer step-by-step, thank you:))

Answer 1

Explanation:

We know a maximum point on the height vs. time curve is at t=16 seconds. Then the height function can be written by filling in the known values in ...

h(t) = (center height) + (wheel radius)·cos((frequency)·2π·(t -(time at max height)))

Since t is in seconds, we want the frequency in revolutions per second. That will be ...

(3.2 rev/min)·(1 min)/(60 sec) = 3.2/60 rev/sec = 4/75 rev/sec

Then our height function is ...

h(t) = 59 + 45·cos(8π/75·(t -16))

9 minutes is 9·60 sec = 540 sec, so we want to find the value of h(540).

h(540) = 59 + 45·cos(8π/75·(540 -16))

= 59 +45·cos(4192π/75)

≈ 59 + 45·0.944376 . . . . . calculator in radians mode

≈ 101.496937 . . . . feet

_____

The cosine function is a maximum when its argument is zero. We used the process of function translation to translate the maximum point to t=16 from t=0. That is, we replaced t in the usual cosine function with (t-16).

We can also evaluate the cosine function by subtracting multiples of 2π from the argument. When we do that, we find that Shirley's height at 9 minutes is the same as it is after 15 seconds. Some calculators evaluate smaller cosine arguments more accurately than they do larger argument values.