Gravel is being dumped from a conveyor belt at a rate of ft /min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is ft high? Round the result to the nearest hundredth.

Question

Gravel is being dumped from a conveyor belt at a rate of  ft /min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is  ft high? Round the result to the nearest hundredth.

Quizplus · Accepted Answer

Let's call the radius and height of the cone "r" and "h" respectively.

Since the base diameter and height are always equal, we know that the cone is a right cone. This means that we can use the Pythagorean theorem to relate the radius and height:

r^2 + h^2 = (2r)^2

Simplifying this equation, we get:

h^2 = 3r^2

Taking the derivative of both sides with respect to time, t:

2h (dh/dt) = 6r (dr/dt)

Now we need to find values for h and r at the given height of the pile, which is h = 11.

To find r, we can use the fact that the base diameter is equal to the height:

2r = h = 11 => r = 5.5

To find dr/dt, we need to relate it to the given rate of gravel being dumped:

We know that the volume of a cone is given by:

V = (1/3)πr^2h

Taking the derivative of both sides with respect to time, t:

dV/dt = (1/3)(2πrh)(dr/dt) + (1/3)πr^2(dh/dt)

We can simplify this using the fact that the rate of change of volume is equal to the rate of gravel being dumped:

11eaa3e5_55e5_718c_9f8f_d1556e197c6b_TB5971_11 = (1/3)(2πr(5.5))(dr/dt) + (1/3)π(5.5)^2(dh/dt)

11eaa3e5_55e5_718d_9f8f_0921dddcae9e_TB5971_11 = (11/3)π(dr/dt) + (15.17)(dh/dt)

Now we can solve for dh/dt:

dh/dt = (1/15.17)(11eaa3e5_55e5_718d_9f8f_0921dddcae9e_TB5971_11 - (11/3)π(dr/dt))

Plugging in the values we know:

dh/dt = (1/15.17)(11eaa3e5_55e5_718d_9f8f_0921dddcae9e_TB5971_11 - (11/3)π(5.5)(11eaa3e5_55e5_72f5_9f8f_3178597975a4_TB5971_11))

dh/dt ≈ 11eaa3e5_55e5_9a09_9f8f_e5a5df810d16_TB5971_11

Rounding to the nearest hundredth, we get:

dh/dt ≈ 11eaa3e5_55e5_9a09_9f8f_e5a5df810d16_TB5971_00

Gravel Is Being Dumped from a Conveyor Belt at a Rate