Sand is being poured from a converyer belt forming a conical pile whose base diameter is eual to its height at all times. If the base diameter is inreasing at 2 m/min when the base is 2 m wide. How fast is the sand being poured onto the pile?

given: 2r = h

d(diameter)/dt = 2 m/min
d(2r)/dt = 2 m/min
d(r)/dt = 1 m/min

V(ofcone) = (1/3)πr^2 h
= (1/3)πr^2 (2r)
= (2/3)π r^3

dV/dt = 2πr^2 dr/dt
= 2π(1^2)((1)

= 2π m^3/min

To find out how fast the sand is being poured onto the pile, we need to determine the rate at which the height of the pile is increasing.

Let's assume that at any given time, the height of the pile is "h" meters and the base diameter is "d" meters. Given that the base diameter is equal to the height at all times, we have:

d = h

Now, we are given that the base diameter is increasing at a rate of 2 m/min when the base is 2 m wide. So, we can express the rate of change of the diameter as:

d'(t) = 2 m/min

To find the rate at which the height is increasing, we need to differentiate both sides of the equation d = h with respect to time (t):

d/dt [d] = d/dt [h]

Using the chain rule, the left side can be written as:

d/dt [d] = d/dt [h] + d/dh [h] * dh/dt

As d = h, we can simplify the equation to:

d/dt [h] = d/dh [h] * dh/dt

Now, let's calculate the derivative d/dh [h]. Taking the derivative of h with respect to h, we get:

d/dh [h] = 1

Substituting this back into our equation, we have:

d/dt [h] = 1 * dh/dt

Therefore, the rate at which the height of the pile is increasing is equal to the rate at which the base diameter is increasing.

Now, we are given that the base diameter is increasing at a rate of 2 m/min when the base is 2 m wide. This means that dh/dt = 2 m/min.

To summarize:

- The rate at which the height of the pile is increasing (d/dt [h]) is equal to the rate at which the base diameter is increasing (d'(t)), which is 2 m/min.

So, the sand is being poured onto the pile at a rate of 2 m/min.