SOLUTION: 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 deno

Question 1048339: 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 by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
i) arc length(in radians) = theta * radus(r)
:
theta = arc length / r
:
use implicit differentiation
:
d(theta)/ dt = d(arc length)/ dt * (1/12)
:
note that d(arc length) /dt = 4 cm/sec
:
d(theta/dt = 4 /12 = 1/3
:
*******************************************************
theta increases at the rate of (1/3) radians per second
*******************************************************
:
ii) area of sector (in radians) = (theta/2) * r^2
:
use implicit differentiation
:
d(area of sector)/dt = d(theta)/dt * (2r/2)
:
d(area of sector)/dt = d(theta)/dt * r
:
d(area of sector)/dt = (1/3) * 12 = 4
:
*************************************************************
area of sector in radians is increasing at 4 radians / second
*************************************************************
:

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