Wheat is poured through a chute at the rate of 20 feet cubed per minute in a con
ID: 2859604 • Letter: W
Question
Wheat is poured through a chute at the rate of 20 feet cubed per minute in a conical pile whose bottom radius is always half the altitude. (i) How fast will the are of the base be changing when the pile is 6 Feet high? (ii) How fast will the circumference of the base be changing when the pile is 6 feet high? Recall, the volume of a cone is given by the formula: V=1/3pir^2h.(5) The sides of a square baseball diamond are 90 feet long, When a player who is between second and third base is 60 feet from second base and heading towards third base at a speed of 22 feet per second, how fast is the distance between the player and home plate changing?Explanation / Answer
volume v =(1/3)pi r2 h
given radius is half the altitude
r=(1/2)h
=>h=2r
volume v =(1/3)pi r2 (2r)
volume v =(2/3)pi r3
differentiate with respect to time t
dv/dt =(2/3)pi 3r2dr/dt
dv/dt =2pi r2dr/dt
when height is 6ft , radius of base =3ft,wheat pored at 20ft3/min
r =3, dv/dt =20ft3/min
20=2pi*32dr/dt
dr/dt =20/(18pi)
a)area of base A=pi r2
differentiate with respect to time t
dA/dt =2pi r dr/dt
dA/dt =2*pi*3*20/(18pi)
dA/dt =20/3
dA/dt =6.67
area of base increasing at 6.67ft2/min
b)circumference of base C=2pi r
differentiate with respect to time t
dC/dt =2pi dr/dt
dC/dt =2pi*20/(18pi)
dC/dt =20/9=2.22
circumference increasing at 2.22 ft/min
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