Rates of Change

Question

N is a fixed point on the circumference of a circle, centre O,radius 8 cm. A variable point P moves around the circumference such that θ (or ∠PON) increases at a constant rate of π/2 rad. per second.Find

 

(a) The rate of increase of s (arc length from N to P)   Ans: 4π cm s^-1

(b) The rate of increase of A, the area of sector NOP   Ans: 16π cm^2 s^-1

 

Answer 1

We will use the equation for arc length.

 

s = rθ 

 

a)Take a derivative on both sides with respect to time:

 

ds/dt = r*dθ/dt = (8 cm)(.5π/s) = 4π cm/s

 

b) A = πr² for a full circle

 

For a partial circle:

 

A = πr²*θ/2π = θ*r²/2

 

dA/dt = rθ*dr/dt + (r²/2)(dθ/dt)

 

dr/dt = 0 since the radius is constant

 

dA/dt = (r²/2)(dθ/dt) = (64 cm²/2)(π/2 rad/s) = 16π cm²/sec