# High School: Geometry

### Modeling with Geometry HSG-MG.A.2

2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

How many monkeys can fit in a barrel? How many slams are there in an old screen door? How many licks does it take to get to the Tootsie Roll center of a Tootsie Pop?

If your students are packed into your classroom like sardines, they may begin to question the school board's answer to, "How many students can we squeeze into a classroom?" But they won't make much ground complaining about the lack of elbowroom. Instead, they can reflect on the population density of their state compared with the neighbors, the salinity of the aquarium in the library, or how nutrient-dense the soil is in the bean field across the street.

Students should understand that density is a ratio of mass (or heat or people or things) to area or volume. The real world boasts many situations modeled by density concepts, but understanding doesn't come from chucking objects into water to see whether they sink or float. Instead, students can take a look at all the different examples of density all around us.

• Ohio asks its mail carriers to help count cottontail rabbits in order to determine the population density of that species of wildlife throughout the state. No lie.
• The USDA provides reports about the national yield of bushels of corn harvested per acre planted.
• Geologists record the quantity of volcanic eruptions (or other rude awakenings) that occurs in the Ring of Fire. Er, uh…we meant the Pacific Ring of Fire.

Like the professor trying to inspire his students to kick back and enjoy a few cold, uh, Yoohoo's with friends, students should try to come up with their own examples of how many things can fit into different spaces and the implications of those densities.

Students must also be able to use density to calculate other quantities related to it and interpret these answers in terms of their contexts. Using the right units, among other things, is good way to check that an answer makes sense.

If we're looking for mass, our answer should be in units of grams or pounds. If we're looking for the number of licks it takes to get to the center of a Tootsie Pop, we should have units of licks per Tootsie Pop. What is that number? The world may never know.

#### Drills

1. A hot air balloon holds 74,000 cubic meters of helium, a very noble gas with the density of 0.1785 kilograms per cubic meter. How many kilograms of helium does the balloon contain?

13,209 kg

Density is calculated as , or mass divided by volume. Fill in the two parts we know and solve for m = Vd = 74,000 m3 × 0.1785 kgm3 = 13,209 kg.

2. The population density of Themville is 17.5 Thems per acre. Exactly 840 Thems live in Themville. How many acres is Themville?

48 acres

Using , we can plug in the data we know to get 17.5 . Solve for V = 48 acres. Of course, when talking about population density, we're really using more of an area measurement than a volume one, but the formula holds true either way.

3. A Wisconsin-based dairy farm has 1,200 cows in their milking herd that collectively produce 160,500 pounds of milk per square kilometer of grazing pasture, for a total of 7,704,000 pounds of milk products. How many square kilometers do these cows get to roam around on?

48 km2

We can ignore the number of cows for this question. That's not the piece of information we care about. To find the number of square kilometers grazed, we divide the number of pounds of milk by the milk per square kilometer, or .

4. Twenty-seven people skated their hearts out at big Ben's birthday party at the skating rink last night, each producing an amazing 2,400 BTUs (British Thermal Units) of heat while listening to an hour's worth of Beatles, Spice Girls, and Adele. The interior of the skating rink is 90 feet long, 40 feet wide, and 20 feet tall. To the nearest thousandth, how many BTUs per cubic foot did they produce?

0.900 BTUs/ft3

We know the total amount of heat produced was 2,400 BTUs × 27 = 64,800 BTUs, and the total cubic feet in the rink is 90 ft × 40 ft × 20 ft = 72,000 ft3. The BTUs per cubic foot is just 64,800 BTUs divided by 72,000 ft3 = 0.9 BTUs per cubic foot.

5. Neil Glennbuzz, astronaut extraordinaire, has returned to the Kennedy Space Center with samples from his recent interplanetary travels. He has 3.47 kilograms of soil from Mars. Given that Mars has an average soil density of 3.93 g/cm3, how many cubic centimeters of soil is that?

882.952 cm3

Given the density of Mars is 3.93 g/cm3, we can solve the density formula for V in cubic centimeters. If we substitute in our density and mass values (remembering to convert 3.47 kg into 3,470 g), we'll be able to solve V for 882.952 cm3.

6. Over a 24-hour period, brown bears counted 840 salmon swimming upstream, and they safely assumed that they only counted 30% of the total number of fish going by. The salmon-spawning haven along Copper River measures approximately 90,000 cubic meters of water. Given that the haven was empty before this week, that the salmon swim upstream at a constant rate, and that once they reach the haven, the salmon hang out there indefinitely, what will the population density of salmon in the Copper River spawning haven be after one week (to the nearest thousandth fish/m3)?

0.218 fish/m3

The bears only counted 30%, which means the salmon are swimming upstream at a rate of 2,800 fish per day, or 19,600 fish per week. The population density will be .

7. The Mom & Pop Coffee Shop wants to open new locations, either downtown or uptown. They will open a new location wherever the ratio of existing coffee shops per person is less than 0.01. The population density of the 20-city-block downtown area is 225 people per city block, but there are already 48 coffee shops in the area. The population of the 30-block uptown area is 125 people per block, and there are 16 coffee shops around. Where, if anywhere, should they open their new location(s)?

Uptown only

First, we should find the total number of people in the uptown and downtown areas. The 225 people per block × 20 blocks downtown = 4,500 people downtown, while the uptown area has 125 people per block × 30 blocks uptown = 3,750 people uptown. Now, we can divide these by the number of coffee shops to get the ratios we need. Downtown, we have a ratio of , but uptown, the ratio is only . They can open a new uptown shop because it's less than the specified 0.01 ratio.

8. In the 1980 census, the population density of a given 15 km2 region was 19.8 people per square kilometer. This density grew to 26.2 people per square kilometer by 1990. In 2000, the population density of the area was 36.4 people per square kilometer. And in 2010, there were 47.6 people per square kilometer. Which statement best describes the population growth of the area?

The biggest increase in number of residents in the area occurred between 2000 and 2010.

Using the population densities and the total area of the region, we can create the following table to show how the population grew each decade.

 Population density Number of People Change in Population % Change in population 1980 19.8 297 1990 26.2 393 96 32.3% 2000 36.4 546 153 38.9% 2010 47.5 714 168 30.7%

Of the answer choices, only (D) is correct. While its percentage wasn't the greatest, the 10 years between 2000 and 2010 did see the highest number of residents.

9. Farmers in Field County are experimenting with new varieties of corn to see if they can increase their crop yield per acre. They planted 452 acres with Lotsacorn and collected a total of 69,110 bushels. They also planted 396 acres with Cornaplenty, which yielded a total of 59,637 bushels. Which of these statements most accurately describes the experiment?

Lotsacorn yielded 1.5% more per acre than Cornaplenty.

Lotsacorn gave a yield of , while Cornaplenty gave a yield of . This means Lotsacorn boasted a 1.5% larger return per acre than Cornaplenty.

10. A balloon archway has been ordered for decorating the high school gym for a 1950's style prom. The archway will contain 275 balloons, each holding 1.2 liters of helium. Each balloon has a mass of 27 milligrams when empty, and all the string and fixings that hold the balloons together total another 45 grams. If helium has a density of 0.1785 grams per liter, what is the mass of the entire archway?