Question

Water is poured into a bowl at a constant rate of 17.0 cm^3/s. The bowl has a circular cross section, but does not have a uniform diameter. (That is, different horizontal cross sections taken at different heights of the bowl have different diameters.) As the water fills the bowl, the water level reaches a point where the diameter of the bowl is

d1 = 1.45 cm.
What is the rate (in cm/s) at which the water level rises at this diameter?

Answer 1

The rate at which the water level rises at a diameter of 1.45 cm is 8.44 cm/s.

Step-by-step explanation:

To find the rate at which the water level rises at this diameter, we can use the concept of continuity equation. The continuity equation states that the flow rate of a fluid is constant at all points in a pipe or container.

Therefore, the rate at which the water level rises at a particular diameter can be found by dividing the flow rate by the cross-sectional area at that diameter.

Rate of water level rise = Flow rate / Cross-sectional area

Substituting the given values, we get:

Rate of water level rise = 17.0 cm³/s / (π * (d1/2)²)

Rate of water level rise = 17.0 cm³/s / (π * (1.45 cm/2)²)

Rate of water level rise = 8.44 cm/s

Answer 2

10.29 cm/s

Step-by-step explanation:

Discharge in to the bowl = 17.0 cm³/s

Diameter of the bowl, d₁ = 1.45 cm

Now,

Rate at which water level rise at its diameter =
`\frac{\textup{Discharge}}{\textup{Area of cross-section}}`

also,

Area of cross-section =
`\frac{\pi}{\textup{4}}*1.45^2`

or

Area of cross-section = 1.651 cm²

Therefore,

Rate at which water level rise at its diameter =
`\frac{\textup{17}}{\textup{1.651}}`

or

Rate at which water level rise at its diameter = 10.29 cm/s