ce amath---rate of change

8. Suppose the origin is the starting point.
Let (x, 0) be the coordinates of the car travelling due east at time t.
Let (0, y) be the coordinates of the car travelling due north at time t.
From the given information, we have
dx/dt = 80,
dy/dt = 60, and after 3 hours,
x = (80)(3) = 240
y = (60)(3) = 180
Let s be the distance between the two cars at time t. Then
s² = x² + y²
Differentiating both sides with respect to t, we have
2s(ds/dt) = 2x(dx/dt) + 2y(dy/dt)
After 3 hours, s² = 240² + 180², s = 300
So, 2(300)(ds/dt) = 2(240)(80) + 2(180)(60)
ds/dt = 100
Therefore, the cars are separating at a rate of 100km/h after 3 hours.

9. Let r cm, S cm and A cm² be the radius, the circumference and the area of the circle respectively. Then
S = 2πr and A = πr²
dS/dt = 2π(dr/dt) ......(1) and dA/dt = 2πr(dr/dt) ......(2)
From the given information, dA/dt = -6
When r = 3, from (2), we have
-6 = 2π(3)(dr/dt)
dr/dt = -1/π
Then by (1), dS/dt = 2π(-1/π) = -2
Therefore, the circumference is decreasing (changing) at a rate of 2cm/s (-2cm/s) when the radius is 3 cm.