Calculating Water Volume in a Cumulus Cloud

[jav6454]

Homework Statement

A cubic centimeter in a typical cumulus cloud contains 50 to 500 water drops, which have a typical radius of 10 µm. For that range, give the lower value and the higher value, respectively, for the following.

How many cubic meters of water are in a cylindrical cumulus cloud of height 3.2 km and radius 1.2 km?

Homework Equations

Volume Formula of a Cylinder ==>
V = 3.14 * h * r2 (squared)
Volume Formula of a Sphere ==>
V = 4/3 * 3.14 * r3 (cubic)

The Attempt at a Solution

I tried to find the volume of the Cubic Centimeter using the 10µm of a drop and multiplying respectively (50 or 500). However I converted the µm to cm to get the Volume in cm3. Then I convert that to cubic meter.

Later I find the volume for the cloud (I find this in cubic meter). Multiply that times the volume of the respective min and max values for the volume of cubic centimeter in the cloud, but in come sout wrong...what am I doing wrong?

 

[dlgoff]

Without actually seeing each step, I'm guessing that you may be having a conversion probelm from um to m, etc. Other than that, it looks like the logic is Okay.

 

[Moetasim]

Your approach is on the right track, but there are a few things to note:

1. Make sure you are using the correct units throughout your calculations. The radius of the water drops is given in micrometers (µm), so you should use that unit for all calculations involving the drops. When converting to cubic centimeters, you should use the conversion factor of 1 cm = 10,000 µm.

2. When calculating the volume of a cylinder, the formula you have used is for a disk, not a cylinder. The correct formula for a cylinder is V = π * r^2 * h, where r is the radius and h is the height.

3. When calculating the volume of a sphere, the formula you have used is correct, but make sure you are using the correct radius. The radius of a water drop is given as 10 µm, so when using the formula, you should use 0.00001 m as the radius.

Using these corrections, the correct approach would be:

1. Convert the radius of the water drops from 10 µm to meters: 0.00001 m.

2. Calculate the volume of a cubic centimeter using the correct formula for a sphere: V = (4/3) * π * (0.00001)^3 = 4.19 x 10^-15 m^3.

3. Convert the volume of a cubic centimeter to cubic meters: 4.19 x 10^-15 m^3 = 4.19 x 10^-18 m^3.

4. Calculate the volume of the cloud using the correct formula for a cylinder: V = π * (1.2 km)^2 * 3.2 km = 14.51 x 10^9 m^3.

5. Multiply the volume of the cloud by the lower and higher values for the volume of a cubic centimeter (50 and 500, respectively): 14.51 x 10^9 m^3 * 50 = 7.26 x 10^-8 m^3 (lower value) and 14.51 x 10^9 m^3 * 500 = 7.26 x 10^-6 m^3 (higher value).

Therefore, the lower and higher values for the volume of water in the cylindrical cumulus cloud would be 7.26 x 10^-8 m^3 and 7.