Question

sand is pouring from a pipe at the rate of 12cm³/ sec . the falling sand forms a cone on the ground in such a waythat the height of the cone is always 16 th of the radius of the base. how fast is the height of the sand coneincreasing when the height is 4cm ?

Answer 1

The height of the sand cone is increasing at a rate of 192 cm³/sec when the height is 4 cm.

Step-by-step explanation:

Let's denote the height of the sand cone as h and the radius of the base as r.

Given that the height of the cone is always 16th of the radius of the base, we can write the equation h = (16/1) * r = 16r.

To find how fast the height of the sand cone is increasing, we need to differentiate the equation h = 16r with respect to time t.

Using the chain rule, we get dh/dt = 16(dr/dt).

Hence, the rate at which the height is increasing is 16 times the rate at which the radius is increasing.

We are given that dr/dt = 12 cm³/sec, so substituting this value into the equation, we get

dh/dt = 16 * 12

= 192 cm³/sec.

Therefore, the height of the sand cone is increasing at a rate of 192 cm³/sec when the height is 4 cm.