Sand falls from a hopper at a rate of 0.6 cubic meters per hour and forms a coni
ID: 2865460 • Letter: S
Question
Sand falls from a hopper at a rate of 0.6 cubic meters per hour and forms a conical pile beneath. Suppose the radius of the cone is always half the height of the cone. Find the rate at which the radius of the cone increases when the radius is 4 meters. meters per hour Find the rate at which the height of the cone increases when the radius is 4 meters. meters per hour A hemispherical bowl of radius 19 cm contains water to a depth of h cm as shown below. Find the square of the radius of the surface of the water as a function of h. r^2 = The water level drops at a rate of 0.1 cm per hour. At what rate is the radius of the water decreasing when the depth is 2 cm? Round your answer to four decimal places. cm/hrExplanation / Answer
a)given radius of cone is halh of the height of cone
=>r=h/2
=>h=2r
volume of cone V=(1/3)r2h
volume of cone V=(1/3)r2(2r)
volume of cone V=(2/3)r3
differentiate with respect to time t
dV/dt =(2/3)3r2(dr/dt)
dV/dt =2r2(dr/dt)
given dv/dt =0.6 ,r =4
0.6 =242(dr/dt)
0.6=32(dr/dt)
dr/dt =0.6/(32)
dr/dt=0.00597
radius increasing at 0.00597 meters/hour
b)
given radius of cone is halh of the height of cone
=>r=h/2
when r =4 => h=8
volume of cone V=(1/3)r2h
volume of cone V=(1/3)(h/2)2h
volume of cone V=(1/12)h3
differentiate with respect to time t
dV/dt=(1/12)3h2(dh/dt)
dV/dt=(1/4)h2(dh/dt)
dv/dt =0.6, h =8
0.6=(1/4)82(dh/dt)
dh/dt =0.6/(16)
dh/dt=0.01194
height increasing at 0.01194 meters per hour
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