Question

The rate of change of the volume V of water in a tank with respect to time t is directly proportional to the cubed root of the volume. Write a differential equation that describes the relationship. Display keyboard shortcuts for Rich Content Editor

Answer 1

The differential equation becomes -

`(dV)/(dt) = k\sqrt[3]{V}` i.e.
`(dV)/(dt) = kV^{(1)/(3) }`

Explanation:

Given - The rate of change of the volume V of water in a tank with respect to time t is directly proportional to the cubed root of the volume.

To find - Write a differential equation that describes the relationship.

Proof -

Rate of change of volume V with respect to time t is represented by
`(dV)/(dt)`

Now,

Given that,

The rate of change of the volume V of water in a tank with respect to time t is directly proportional to the cubed root of the volume.

⇒
`(dV)/(dt)` ∝
`\sqrt[3]{V}`

Now,

We know that, when we have to remove the Proportionality sign , we just put a constant sign.

Let k be any constant.

So,

The differential equation becomes -

`(dV)/(dt) = k\sqrt[3]{V}` i.e.
`(dV)/(dt) = kV^{(1)/(3) }`