Question

An open storage box shaped like a square prism but without a top face.The base of the box is a square with side length 12 in., and the height of the box is 10 in. What is the surface area of the box.

Answer 1

Therefore the rate change of height is
`(1)/(\pi)` m/s.

Step-by-step explanation:

Given that a vertical cylinder is leaking water at rate of 4 m³/s.

It means the rate change of volume is 4 m³/s.

`(dV)/(dt)=4 \ m^3/s`

The radius of the cylinder remains constant with respect to time, but the height of the water label changes with respect to time.

The height of the cylinder be h(say).

The volume of a cylinder is
`V=\pi r^2 h`

`=(\pi * 2^2 * h) m^3`

`\therefore V=4\pi h`

Differentiating with respect to t.

`(dV)/(dt)= 4\pi (dh)/(dt)`

Putting the value
`(dV)/(dt)`

`4= 4\pi (dh)/(dt)`

`\Rightarrow (dh)/(dt)= (4)/(4\pi)`

`\Rightarrow (dh)/(dt)= (1)/(\pi)`

The rate change of height does not depend on the height.

Therefore the rate change of height is
`(1)/(\pi)` m/s.