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# Density Problems

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A balloon and a rock each have a volume of 1 cubic foot. The rock is much heavier than the balloon. What does this say about the density of the balloon compared to the density of the rock?

#### Guidance

Density is a ratio of two measurements of an object. In the past you might have seen the following formula for density:

$d=\frac{m}{V}$

This formula states that the density of an object is the ratio of its mass to its volume. If two objects have the same volume, but different masses, the object with the greater mass will be denser.

You can also think about the density of an area of land such as a city. In this case, you would look at the ratio of number of people to area of land.

$population \ density=\frac{number \ of \ people}{area \ of \ land}$

A city will have a higher population density than land in the country, because more people live in each square mile of a city than live in each square mile out in the country.

In general, density is a ratio of the mass/people/things to volume/area. Given the density and one of the two quantities in the ratio, you can solve for the other quantity in the ratio.

Example A

Different metals have different densities. The density of gold is 19.3 grams per $cm^3$ , the density of copper is 8.86 grams per $cm^3$ , and the density of bronze is 9.87 grams per $cm^3$ . Your bracelet is 106 grams and has a volume of 12 cubic centimeters. What metal is your bracelet made out of?

Solution: The density of your bracelet is $d=\frac{106}{12} \approx 8.83 \ grams \ per \ cm^3$ . It is probably copper.

Example B

There are approximately 17,620 people per square mile in New York City. If New York City is 468 square miles, approximately how many people live in New York City?

Solution: 17,620 is the population density of New York City.

$17, 620 \ people \ per \ square \ mile=\frac{\# \ of \ people}{468 \ mi^2}$

Solve this equation by multiplying both sides by 468. New York City has approximately 8,246,160 people.

Example C

Helium is a gas lighter than air that is used in balloons. The density of helium is approximately 0.18 kilograms per cubic meter. You get 16 helium balloons for your 16th birthday, each with a volume of 15 cubic meters. How many kilograms of helium do the balloons contain?

Solution: The 16 balloons together have a volume of $16(0.15)=2.4 \ m^3$ . Each cubic meter has a mass of 0.18 kg , so the balloons contain $(2.4)(0.18)=0.432 \ kilograms \ of \ helium$ .

Concept Problem Revisited

A balloon and a rock each have a volume of 1 cubic foot. The rock is much heavier than the balloon. What does this say about the density of the balloon compared to the density of the rock?

Because the balloon and the rock each have the same volume, the object with the greater mass has the greater density. Therefore, the rock is more dense than the balloon.

#### Vocabulary

Density is a ratio of the mass/people/things to volume/area.

#### Guided Practice

A cylindrical candle with a radius of 3 centimeters and a height of 8 centimeters has a mass of 300 grams.

1. What is the density of the candle?

2. Another candle made out of the same wax is formed into a cube. The mass of this candle is 475 grams. What are the dimensions of the cube?

3. Another candle made out of the same wax is formed into a cone. The diameter of the base of the cone is 4 centimeters and the height of the cone is 12 centimeters. What is the mass of the cone candle?

1. $V=\pi r^2 h=\pi (3^2)(8)=226.2 \ cm^3$$d=\frac{300}{226.2}=1.33 \ grams \ per \ cubic \ centimeter$ .

2. $d=\frac{m}{V}$  so $1.33 = \frac{475}{V}$ . Solve this equation and you find that $V=357.14 \ cm^3$ . Because the candle is a cube, all of its edges must be the same length. If $s$  is the length of each edge of the cube, $357.14=s^3$  and $s \approx 7.1 \ cm$ .

3. $V=\frac{\pi r^2 h}{3}=\frac{\pi (2^2)(12)}{3} \approx 50.27 \ cm^3$$d=\frac{m}{V}$ so $1.33=\frac{m}{50.27}$ . Solve this equation and you find that the mass is approximately 66.85 grams.

#### Practice

1. 30 students return to the classroom after playing outside at recess. Over the course of the next hour, each person produces 800 BTUs (British Thermal Units). The room is 20 feet by 20 feet by 12 feet. How many BTUs per cubic foot were produced?

At $20^\circ C$ , skim milk has a density 1.033 kilograms per liter, while light cream has a density of 1.012 kilograms per liter.

2. At $20^\circ C$ , what is the mass of one gallon of skim milk if 1 liter is approximately 0.264 gallons?

3. At $20^\circ C$ , a container holding 9 liters of liquid is 9.108 kilograms. Does the container hold skim milk or light cream?

In Manhattan, there are 15 CoffeeStops per 100,000 people.

4. Manhattan has approximately 1,602,000 people. How many CoffeeStops are there?

5. Manhattan is approximately 34 square miles. What is the density of CoffeeStops per square mile?

6. On average, each person in Manhattan goes to a CoffeeStop three times a week to buy a cup of coffee. About how many cups of coffee does each Manhattan CoffeeStop serve each day?

The population density of a certain 750 square mile area in 1980 was 1077 people per square mile. In 1990, the population density was 1137 people per square mile. In 2000, the population density was 1144 people per square mile. In 2010, the population density was 1193 people per square mile.

7. What was the population of this area in 1980?

8. How much did the population change between 1980 and 2010?

The density of gold is $19.3 \ grams \ per \ cm^3$

10. What is the mass of a gold block that measures 10 cm by 3 cm by 3 cm?

11. A pyramid shaped object is made out of pure gold. The base of the pyramid is a square and the height of the pyramid is 5 centimeters. The mass of the pyramid is 1.5 kilograms. What are the dimensions of the base of the pyramid?

In general, if one object has a lower density than a liquid, it will float in the liquid.

12. At $4^\circ C$ , water has a density of $1000 \ kg/m^3$ . The density of ice is $917 \ kg/m^3$ . Why does ice float in water?

13. A piece of wood has a mass of 25 kg and a volume of $0.000385 \ m^3$ . What is the density of the wood?

14. Do you think the wood from #13 will float in water?

15. A cylinder of lead has a diameter of 3 centimeters and a height of 8 centimeters. The density of lead is $11.4 \ g/cm^3$ . What is the mass of the cylinder?