Related Rates of Change

Question

N is a fixed point on the circumference of a circle, centre O, radius 12cm.

A variable point moves round the circumference with a constant speed of 4cm/s. The angle PON is denoted by θ.

Find
(i) the rate of increase of θ in radians per seconds.
(ii)the rate of increase of the area of the sector NOP.

Answer 1

s/12 = θ.  dθ/dt = (1/12)(ds/dt) = (1/12)(4 cm/sec) = 1/3 rad/sec.

 

A = ½(r²)θ.  dA/dt = ½(r²)(dθ/dt) = 72(1/3) = 24 cm²/sec.