As a Ferris wheel rotates, the height, h, meters, above the ground of one of its cars, can be modeled by the function h(t)=10sin {pi (t-o.5)} + 15, where t is time, in minutes. At what value of time is the instantaneous rate of change the greatest positive value ?
1. T= 0.5 minutes
2. T= 1 minutes
3. T= 0 minutes
4. T= 1.5 minutes
sinθ has its maximum rate of change when θ=0 (that is, when cosθ = 1)
so, you want π(t-0.5) = 0
t = 0.5
The instantaneous rate of change is represented by the first derivative.
To find where that instantaneous rate of change is a maximum we set the 2nd derivative equal to zero.
h(t)=10sin {pi (t-o.5)} + 15
h(t)= 10sin (π(t - 0.5)) + 15
h ' (t) = 10cos(π(t - 0.5)*π
= 10πcos(π(t - 0.5)
h '' (t) = -10πsin(π(t - 0.5)*π
= -10π^2 sin(π(t - 0.5)
-10π^2 sin(π(t - 0.5) = 0
sin(π(t - 0.5) = 0
π(t - 0.5) = 0 or π(t - 0.5) = π/2
t - 0.5 = 0 or t-.5 = 1/2
t = 0.5 OR t = 1
at t = .5, h'(.5) = 10πcos(π(.5 - 0.5) = 10π
at t = 1, h'(1) = 10πcos(π(1 - 0.5) = 0
which value of t gave us the max and which gave us the minimum velocity?
To determine at what value of time the instantaneous rate of change is the greatest positive value, we need to find the derivative of the function h(t) and evaluate it.
Step 1: Find the derivative of h(t)
The derivative of h(t) with respect to t can be found using the chain rule. The derivative of sin(x) is cos(x), and the derivative of a constant (15) is 0.
h'(t) = 10 * cos(π(t-0.5)) * π * 1
Simplifying h'(t), we end up with:
h'(t) = 10πcos(π(t-0.5))
Step 2: Evaluate h'(t) at different values of t
Let's evaluate h'(t) at the given values of t:
For t = 0.5 minutes:
h'(0.5) = 10πcos(π(0.5-0.5)) = 10πcos(0) = 10π(1) = 10π
For t = 1 minutes:
h'(1) = 10πcos(π(1-0.5)) = 10πcos(π/2) = 10π(0) = 0
For t = 0 minutes:
h'(0) = 10πcos(π(0-0.5)) = 10πcos(-π/2) = 10π(0) = 0
For t = 1.5 minutes:
h'(1.5) = 10πcos(π(1.5-0.5)) = 10πcos(π) = 10π(-1) = -10π
Step 3: Compare the values of h'(t)
From the evaluations above, we can see that the values are:
h'(0.5) = 10π
h'(1) = 0
h'(0) = 0
h'(1.5) = -10π
The greatest positive value is 10π, which occurs at t = 0.5 minutes.
Therefore, the correct answer is option 1. T = 0.5 minutes.