Question

9. A swimming pool in the shape of a cylinder has a radius of 2 meters. The water in

the pool is 1 meter deep.
a. The density of water is about 1 gram per cubic centimeter. Find the number
of kilograms of water in the pool.
b. You add 2500 kilograms of water to the pool. What is the depth of the water
in the pool?

Answer 1

The mass of water in the cylindrical pool is approximately 12,566.37 kg, and after adding 2500 kg of water, the new depth is approximately 1.9 meters.

Step-by-step explanation:

Calculating the Mass of Water in a Cylinder:

To find the mass of water in the swimming pool, we first need to determine the volume of water in the pool. Since the pool is a cylinder, we can use the formula for the volume of a cylinder:

V = πr²h,

where

V is the volume,

r is the radius, and

h is the height (or depth in this case).

The volume of water is therefore π * (2 m)² * 1 m = 4π m^3.

Since 1 m^3 = 1,000,000 cm^3 and the density of water is 1 g/cm^3, the mass of the water in kilograms will be:

(4π m^3) * (1,000,000 cm³/m³) * (1 g/cm³) * (1 kg/1000 g) = 4π * 1000 kg = approximately 12,566.37 kg.

Finding the New Depth of the Pool After Adding Water:

After adding 2500 kg of water, the new volume of water in the pool is:

12,566.37 kg + 2500 kg = 15,066.37 kg.

To find the new depth, we divide the total mass of water by the area of the base of the cylinder (which remains unchanged) and by the density of water, and convert from kilograms back to meters cubed:

(15,066.37 kg) / (1 kg/L) / (π * (2 m)^2) = approximately 1.9 meters.

The new depth of the water in the pool is approximately 1.9 meters.

Answer 2

a. To find the number of kilograms of water in the pool, we need to calculate the volume of the pool and then convert it to kilograms.

The formula to calculate the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius, and h is the height (or depth) of the cylinder.

Given:
Radius (r) = 2 meters
Height (h) = 1 meter

Using the formula, the volume of the pool is:
V = π(2^2)(1) = 4π cubic meters

Now, we need to convert the volume from cubic meters to cubic centimeters since the density of water is given in grams per cubic centimeter.

1 cubic meter = 1,000,000 cubic centimeters

So, the volume of the pool in cubic centimeters is:
4π x 1,000,000 = 4,000,000π cubic centimeters

The density of water is 1 gram per cubic centimeter. Therefore, the number of kilograms of water in the pool is:
4,000,000π x 1 = 4,000,000π kilograms

b. If you add 2500 kilograms of water to the pool, we can calculate the new depth of the water.

Using the same formula as before, we can find the new volume of the pool:
V = π(2^2)(h_new) = 4πh_new

We know that the new volume is the previous volume plus the added water:
4πh_new = 4,000,000π + 2500

Simplifying the equation:
h_new = (4,000,000π + 2500) / (4π)

Calculating the value:
h_new ≈ 1,000,000 + 625 / 4 ≈ 250,625 / 4 ≈ 62,656.25 meters

Therefore, the new depth of the water in the pool is approximately 62,656.25 meters.