rates of change (1 Viewer)

[fullonoob]

the angle @ between two radii OP and OQ of a circle of radius 6cm is increasing at the rate of 0.1 radians per minute.
Find the value of @ for which the rate of increase of the area of the segment cut off by the chord PQ is at its maximum.

 

[shaon0]

the angle @ between two radii OP and OQ of a circle of radius 6cm is increasing at the rate of 0.1 radians per minute.
Find the value of @ for which the rate of increase of the area of the segment cut off by the chord PQ is at its maximum.

A{segment}=1/2.r^2.@-(1/2.r^2.sin@)
=(1/2)(r^2)(@-sin@)
dA/d@=(1/2)(r^2)(1-cos@)
dA/dt=dA/d@.d@/dt
=(1/2)(r^2)(1-cos@).(0.1)
=(1/20)(r^2)(1-cos@)
Let dA/dt=0:
cos@=1 => @=0

Idk, if the above is correct as i haven't done rates of change in ages.

 

[fullonoob]

thats what i did but the answer is pi because common sense will say 0 degrees will give no area. But i think i get it now, you absolute value the cos@ to obtain pi

 

[fullonoob]

A rotating light L is situated at sea 180m from the nearest point P on a straight shoreline. The light rotates through one revolution every 10 secs. Show that the rate at which a ray of light moves along the shore at a point 300m from P is 136 pi m/s

 

[jet]

 

[fullonoob]

lol nice, need help on one more but it has a diagram. If you have cambridge maths 3u year 11 textbook its on, ex 14H , q 18 a).
Otherwise, try use this info i give you ><
PQ is a diameter of the circle and S is a point on the circumference. T is the point on PQ such that PS = PT. Let angle SPT = A
show that the area A of triangle SPT is A = .5 d^2 cos^2A sinA, where d is the diameter of the circle. GOOD LUCK