Question

an inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec. Find the rate of change of the water depth when the water depth is 10 ft.

Answer 1

the rate of change of the water depth when the water depth is 10 ft is;

Explanation:

Given that:

the inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec.

We are meant to find the rate of change of the water depth when the water depth is 10 ft.

The diagrammatic expression below clearly interprets the question.

From the image below, assuming h = the depth of the tank at a time t and r = radius of the cone shaped at a time t

Then the similar triangles ΔOCD and ΔOAB is as follows:

( similar triangle property)

h = 2.5r

The volume of the water in the tank is represented by the equation:

The rate of change of the water depth is :

Since the water is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec

Then,

Therefore,

the rate of change of the water at depth h = 10 ft is:

Thus, the rate of change of the water depth when the water depth is 10 ft is;

Explanation:

Answer 2

the rate of change of the water depth when the water depth is 10 ft is;
`\mathbf{(dh)/(dt) = (-25)/(100 \pi) \ \ ft/s}`

Explanation:

Given that:

the inverted conical water tank with a height of 20 ft and a radius of 8 ft is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec.

We are meant to find the rate of change of the water depth when the water depth is 10 ft.

The diagrammatic expression below clearly interprets the question.

From the image below, assuming h = the depth of the tank at a time t and r = radius of the cone shaped at a time t

Then the similar triangles ΔOCD and ΔOAB is as follows:

`(h)/(r)= (20)/(8)` ( similar triangle property)

`(h)/(r)= (5)/(2)`

`(h)/(r)= 2.5`

h = 2.5r

`r = (h)/(2.5)`

The volume of the water in the tank is represented by the equation:

`V = (1)/(3) \pi r^2 h`

`V = (1)/(3) \pi ((h^2)/(6.25)) h`

`V = (1)/(18.75) \pi \ h^3`

The rate of change of the water depth is :

`(dv)/(dt)= (\pi r^2)/(6.25)\ (dh)/(dt)`

Since the water is drained through a hole in the vertex (bottom) at a rate of 4 ft^3/sec

Then,

`(dv)/(dt)= - 4 \ ft^3/sec`

Therefore,

`-4 = (\pi r^2)/(6.25)\ (dh)/(dt)`

the rate of change of the water at depth h = 10 ft is:

`-4 = ( 100 \ \pi )/(6.25)\ (dh)/(dt)`

`100 \pi (dh)/(dt) = -4 * 6.25`

`100 \pi (dh)/(dt) = -25`

`(dh)/(dt) = (-25)/(100 \pi)`

Thus, the rate of change of the water depth when the water depth is 10 ft is;
`\mathtt{(dh)/(dt) = (-25)/(100 \pi) \ \ ft/s}`