Word Problem (use radians)

Question

 

The function h(t) = 25 - 10cos [(π/3) t] gives the height above the ground, in feet, of a passenger on a Ferris wheel t minutes after the ride begins. During one revolution of the Ferris wheel, for how many minutes is the passenger at least 25 feet above the ground?

 

Answer 1

1. First find the points in time where the height ≥ 25 ft
1. h(t) = 25 - 10cos[(π/3) t] = 25
2. -10cos[(π/3) t] =  0  {Sorry about that weak pi symbol.  Subtract 25 from both sides}
3. cos[(π/3) t] = 0   {Divide both sides by -10)
4. (πt/3) = cos^-1(0)
5. The inverse cosine of zero is π/2 or (3π/2).
2. For the first case:  (πt/3) = π/2
1. t = (π/2)(3/π) =  3/2
3. For the second case:  (πt/3) = 3π/2
1. t = (3π/2)(3/π)  =  9/2
4. To find the number of minutes above 25, simply subtract the two times,
1. 9/2 - 3/2
2. 9/2 - 3/2  = 6/2
5. The ferris wheel is above 25 feet for 3 minutes.

Answer 2

h(t) = 25 - 10 cos (π/3)t

let u = (π/3)t

25 ≤ 25 - 10 cos u

0 ≤ - 10 cos u

0 ≤ cos u

This occurs when 0 < u < π

Therefore

0 < (π/3)t < π

0 < πt < 3π

0 < t < 3