The radius r (cm) of a circle at time t seconds is given by r = 9t - t^3 . At each

of the following instants, find the rate of change of the radius (w.r.t.'t') and state
whether the radius is increasing or decreasing at these instants.
a) t =1 b) t =2 c) t =2.5

To find the rate of change of the radius (dr/dt) at a specific instant, we need to differentiate the equation for the radius with respect to time (t). After finding the derivative, we can substitute the given values of time to determine the rate of change and whether the radius is increasing or decreasing.

Given: r = 9t - t^3

First, let's find the derivative of r with respect to t:
dr/dt = d/dt(9t - t^3)

To differentiate, we apply the power rule:
dr/dt = d/dt(9t) - d/dt(t^3)
dr/dt = 9 - 3t^2

Now we can substitute the given values of time to find the rate of change and determine whether the radius is increasing or decreasing:
a) t = 1
dr/dt = 9 - 3(1)^2
dr/dt = 9 - 3
dr/dt = 6

Since the rate of change is positive (6), the radius is increasing at t = 1.

b) t = 2
dr/dt = 9 - 3(2)^2
dr/dt = 9 - 12
dr/dt = -3

Since the rate of change is negative (-3), the radius is decreasing at t = 2.

c) t = 2.5
dr/dt = 9 - 3(2.5)^2
dr/dt = 9 - 18.75
dr/dt = -9.75

Since the rate of change is negative (-9.75), the radius is decreasing at t = 2.5.