Geometry Workbook 13unionparish.enschool.org/ourpages/auto/2018/11/1/70881963/Workbook13.pdfNov 01, 2018 · I will be able to determine and calculate appropriate measures to real

Geometry Workbook 13:

Geometric Modeling and Population Density

Student Name __________________________________________

STANDARDS:

G.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

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

SKILLS:

I will be able to approximate real world objects with geometric shapes and solids.

I will be able to determine and calculate appropriate measures to real world objects using

geometric shapes and solids.

I will be able to calculate density (D), mass (m) or volume (V) using the formula, 𝐷 =𝑚

𝑉 .

I will be able to use area relationships and formulas to solve population density problems.

Notes:

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G.MG.A.1 NOTES – geometrycommoncore 1

Use geometric shapes, their measures, and their properties to describe

objects (e.g., modeling a tree trunk or a human torso as a cylinder).

CONCEPT 1 – Recognize geometric shapes in the objects around us.

RECOGNIZING Modeling is often the process of using something familiar to create a ‘likeness’ of something else. We use models because they are often either cheaper or easier to make. By creating a simple ‘likeness’ to an object or situation we can learn more about it without the complication. Ultimately modeling is a very important skill to have. Geometric shapes (2-D objects) and solids (3-D objects) are great for this because we know a lot them and most things can be reduced to them.

For example, these everyday items can be simplified greatly by describing them as geometric objects.

Dice Bike Tire Barbell Weight Puzzle Bowl

Cubes

(Rectangular Prism) Circles (2-D) Torus (3-D)

Circle (2-D) Cylinder (3-D)

Pyramid (3-D) Hemisphere

Once a shape or solid has been associated to the object you can begin to calculate. Remember that error is introduced when your model doesn’t exactly match the object. An example of this error is found in the table above when we estimated the bowl as a hemisphere – most bowls are much shallower and would not rest on a table if it was a hemisphere. These differences will introduce some element of error. That is normal, modeling is often used as a form of estimating - getting close answers to inform us and not necessarily exact ones. In this environment, a wider variety of answers can be acceptable because we are using the model to understand what is going on. The better the model, of course, the better the calculations. When we create a model, we do our best to have it as close as possible to the actual object. For instance, a hemisphere would be a better model of a baseball cap than a pyramid.

OBJECT GOOD MODEL BAD MODEL

ESTIMATING

You can also use measurements, dimensions and relationships to approximate values about the model. For example, let’s say that last week you bought a stove and its dimensions were 30 in. wide, 36 in. high and 25 in. deep. Could you estimate the size of the dishwasher?

It would be basically the same size as the stove dimensions.


Could you estimate the size of the refrigerator? or even figure out what its volume is.

Of course - it looks to be about twice the height and the same width and depth as the stove. It is probably

30 in. wide, 72 in. high and 25 in. deep. Having the approximate dimensions would allow us to calculate the

volume because our model is a rectangular prism and its volume formulas is V = Bh.

3(30)(72)(25) 54,000V Bh lw h in

DIMENSIONAL ANALYSIS (UNIT CONVERSION)

Part of modeling is also working with lots of types of units. In this case we might want to express our

relationship in terms of cubic feet instead of cubic inches. To make the dimensional change, we need to know

their relationship, 1 foot = 12 inches.

33

3 31 54,000(54,000 ) 31.25

12 1728

ft ftin ft

in

Notice we wanted cubic feet so we cubed the relationship. Dimensional analysis is also a very important part of modeling. Here are a few relationships that you will probably want as a reference.

1 foot = 12 inches 1 mile = 5280 feet 1 pound = 453.592 grams

1 ft3 = 1728 in.3 1 kilometer = 0.621371 mile 1 mile = 1.60934 kilometer

1 ft = 0.000189394 mile 1 foot = 0.3048 meter

ROUNDING

Because estimating and modeling is not necessarily about getting the exact number you will using rounding to

simplify the number. The table below provides a few examples of the rounding process.

Initial Number Location Look to right.. Rounded #

37,549.905 To hundredth 37,549.905 37,549.91

37,549.905 To tenths 37,549.905 37,549.90

37,549.905 To ones (unit) 37,549.905 37,550.00

37,549.905 To tens 37,549.905 37,550.00

37,549.905 To hundreds 37,549.905 37,500.00

37,549.905 To thousands 37,549.905 38,000.00


CONCEPT 2 – Apply geometric measurements and calculations to the models.

Geometric models are generally formed so that you can analyze them by finding the perimeter, surface area,

area or volume of them. I have provided a few of the basic formulas in the tables below as a quick reference

to help you.

Shape Perimeter Area

Rectangle P = 2l + 2w A lw

Square P = 4s 2A s

Parallelogram P = sum all four sides A bh

Triangle P = sum all three sides 1

2A bh

Trapezoid P = sum all four sides 1 2

1( )

2A b b h

Circle 2C r 2A r

Circle Sector 2

360A r

Solid Volume Volume Specific

Rectangular Right Prism V Bh V bh h

Triangular Right Prism V Bh 1

2V bh h

Trapezoidal Right Prism V Bh 1 2

1( )

2V b b h h

Cylinder V Bh 2V r h

Rectangular Pyramid 1

3V Bh

1

3V bh h

Cone 1

3V Bh 21

3V r h

Sphere 34

3V r

Hemisphere 32

3V r

G.MG.A.1 WORSHEET #1 – geometrycommoncore NAME: ______________________ 1

1. Name a geometric shape or solid that would approximate the form of the given object.

a) A tree trunk b) a door c) a plate d) a bowl

________________ ________________ ________________ ________________ e) a pillow f) a tire g) a human leg h) a soccer ball

________________ ________________ ________________ ________________

2. Describe the geometric solids that would best represent these given LAMPSHADES.

a) ______________________ b) ______________________ c) ______________________

d) ______________________ e) ______________________ f) ______________________

3. Draw/Outline the solids (3-D objects) that make up these objects and describe them. a) A house b) A boy c) An airplane

d) A couch e) A pool f) A barbell

G.MG.A.1 WORSHEET #1 – geometrycommoncore 2

4. Determine a geometric shape (2-D) that is approximately the same form as the following objects. a) Wall Decoration (#1) ________________________ b) 3-picture Frame (#2) ________________________ c) Small Picture Fame (#3) ________________________

d) Which do you think would take up more wall space -- three of the Wall Decorations (#1) or one of the

Picture Frames (#2)? Explain.

5. Determine a geometric solid (3-D) that is approximately the same form as the following objects. a) Suit Case (#4) ________________________

b) Lampshade (#5) ________________________

c) Small Pillow (#6) ________________________

d) The bed is a standard double (54 in. by 74 in.) with a single mattress that is 12 inches thick. Approximate the dimensions of the dresser attached to the bed.

e) The dresser by the windows is the same size as the one that you measured in part d. Televisions are

measured by the diagonal length from one corner to the other. What is the approximate size of the TV

pictured?


1. Calculate the approximate volume of this young boy’s torso area (main body region) in two different ways: as a cylinder and as a right prism. If you look at him sideways, he seems to be about 6 inches wide. Which estimate do you like better? Why?

2. The below picture represents a farmer’s market full of vegetables.

a) Fourteen cucumbers that are almost identical in size (16 cm long and 4 cm diameter) are purchased from the market. If the cucumbers are stacked as shown in the diagram, what are the dimensions of the box? Give dimensions to the nearest cm.

b) Thirty-four onions all about the same size fit almost perfectly in the basket, as shown above. There is very

little space that is not filled by the onions. If the basket has a height of 30 cm and a diameter of 30 cm, find

the approximate radius of one onion. (round answer to the tenths place)

c) The volume of the green bean basket is 1.2 times larger than the volume of the onion basket. If the green

beans come right to the top of the basket, how many green beans do you think are in the barrel? The average

green bean is 9 cm with a radius of 0.6 cm. (round answer to the tens place)


3. The little lamp has a cylindrical base with a height of 15 cm and a diameter of 8 cm. Its hemispherical lampshade has a diameter of 14 cm. If each dimensions are 1.5 times bigger than the previous lamp, what is the total volume of the largest lamp? (round answer to the tens place)

4. These lamps are a special design because they work under water. The three identical bases are a heavy cylindrical weight so that they sink. Each base has a height of 4 cm and a diameter of 8 cm. The first two lamps are identical in shape except the second one is truncated (cut off) so that it no longer looks like a cone. The third one looks a lot like half of a sphere. The approximate dimensions are shown below. If these special lamps were placed in a tank of water, what would be the total volume of water they would displace? (round answer to the nearest tens place)


5. A modern wood lamp has two parts: a solid lower piece and an identical upper piece except for the three lights. The depth of the lamp is the same as the diameter of a spherical bulb. What is the volume of the wood used in this lamp? A blueprint with more detailed dimensions is provided to help you with this. (round answer to the nearest tens place). Hint -- look carefully at the shape of the holes they aren’t spherical.


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

CONCEPT 1 – Working with density (Volume)

The basic relation for density is m

DV

, where m is the mass (or weight) and V is volume. Density is defined

as the degree of compactness of a substance. This seems a little unclear so let’s look at an example. In the table below, there are three balls that are the same shape and size but obviously they do not have the same weight or density. The steel ball is very ‘compact’ while the ping pong ball is not ‘compact’ at all – the Steel ball is DENSER than the golf ball or the ping pong ball.

Golf Ball Ping Pong Ball Steel Ball

Like most formulas, you will need to solve for the different variables and so different variants of it will be helpful.

mD

V

mV

D m D V

The mass (or weight) of the object can be measure with many different units. Some of the common ones are pounds (lbs.), grams (g), kilograms (kg) or ounces (oz.) but this is by far not comprehensive. You transition between these measurements using conversion relationships such as these two.

1 pound = 453.592 grams 16 ounces = 1 pound

Conversions will also be made between the units of the volume. Similarity, volume can be measured using lots of different units but some of the common ones will be cubic inches (in3), cubic centimeters (cm3), feet (ft3), cubic meters (m3), cubic miles (mi3) or cubic kilometers (km3). Once again here are a few of the common conversion relationships.

1 foot = 12 inches 1 mile = 5280 feet

1 kilometer = 0.621371 mile 1 mile = 1.60934 kilometer

1 ft = 0.000189394 mile 1 foot = 0.3048 meter

Just remember these conversions are one-dimensional comparisons so when converting cubic feet to cubic inches you need to adjust the relationship as shown.

1 foot = 12 in. (1 foot)3 = (12 in.)3 1 ft3 = 1728 in.3


Here are a few density examples. #1 What is the density of a golf ball, if it has a diameter of 1.66 in. and a weight of 45.63 grams?

33 3

45.63 2.23

4 4(0.83)

3 3

m m gD

V inr

I guess that is why they sink..

#2 Jack has twenty-four ten-foot 2 by 4’s beside his house. He wants to stack them on a shelf but is not sure how much they weigh. If the density of the wood is 35.9 lbs/ft3, determine their weight. (round to the nearest pound)

Step #1 Find the Volume.

Step #2 Convert cubic inches to cubic feet.

Step #3 Determine the weight.

10 ft = 120 inches

3

(2)(4)(120)

960

V Bh

in

3

3 31 960960 0.56

12 . 1728

ftin ft

in

3

3

35.9( ) (0.56 )

20.10

20

lbsm D V ft

ft

lbs

lbs

#3 A hemispherical tank is filled with water and has a diameter of 10 feet. If water weighs 62.4 pounds per cubic foot, what is the total weight of the water in a full tank, to the nearest pound?

Step #1 Find the Volume. Step #2 Determine the weight.

33

3

2 25

3 3

261.80

V r

ft

3

3

62.4261.80

16,336.32

lbsm DV m ft

ft

lbs

CONCEPT 2 – Working with density (Area)

The most common density dealing with area is called Population Density. Population Density is the number of

people (or living organisms) existing per unit of an area (e.g per square mile); also simplified as the number of

people (or organisms) relative to the space occupied by them.

People per square mile Plants per square foot Mice per square kilometer

2

256 people

mi

2

0.54 plants

ft

2

5.88mice

km

Population density informs us about how ‘compact’ the conditions are in a region. For example, look at the

chart below to get an idea of the diversity of leaving conditions across the country.

City Population Density City Population Density

Philadelphia, PA 2

11,233 people

mi

Seattle, WA 2

4,722 people

mi

City Population Density City Population Density

Anchorage, AL 2

175 people

mi

New York, NY 2

27,016 people

mi


Let’s look at some examples of population density.

#1 The Jefferson family own a 100-acre ranch. On the ranch, they have 88 cattle.

What is the population density of the cattle on the ranch in square miles?

Step #1 Convert acres to miles

Step #2 Determine the Population Density.

(1 acre = 0.0015625 mi2)

2

20.0015625100 0.1563

1

miacres mi

acre

2 2

88 563

0.1563

population cattle cattleD

area mi mi

#2 Use the given map and scale to estimate the population density of Utah if its population is 3,051,217.

Starting Map 1.Outline Map (Easy Shapes) 2. Determine scale 3. Measure shapes

4. Convert values to miles 5. Calculate Areas 6. Sum Areas 7. Calculate Pop. Density

2

2

2

3,051,217

81,837.26

37.28

37

peopleD

mi

people

mi

people

mi

.

The actual numbers for this problem are 84,899 square miles and 2

36 peopleD

mi .

#3 A population of 578 honey bees live in a 1-acre area. They gather nectar from a population of 310

flowering plants. The plants live in a 0.2-acre area. Which population has greater density, the insects or the

plants? Why?

Honey Bees Plants Adjust the plants to have the same region

578 beesD

acre

310

0.2

plantsD

acre

310 5 1550

0.2 5

plants plamtsD

acre acre

The plants have the greater population density. They produce much more seed in the region. Also bees life

spand is shorter.


1. Which weighs more -- a pound of feathers or a pound of rocks?

2. Given the following comparisons, circle the one that you think weighs MORE. a) b) c)

3. Jack goes to a hardware store to pick up some 2 x 4’s (planks of wood). He picks up two that are the same size, but one weighs much more than the other. How could that be?

4 a) A logger cuts down a 30-foot tall Beech tree that has a diameter of 4 feet. Trees that are cut for timber are stripped of their branches making them a cylindrical shape. Determine the approximate weight of the tree to the nearest pound. b) Which wood on this list is the best for building a canoe? Is there anything else you might need to know about the wood to help validate your choice? c) Jennifer wants to make a giant block stacking game. To do this she needs to make 54 (6 in. x 3 in. x 2 in.) blocks of wood. What would be the weight difference between using Hickory verses White Pine? Explain which you would use. (1 ft3 = 1728 in3)

Wood Type Density lb/ft3

Beech 44.2

Cherry 36.7

Hickory 50.9

Paper Birch 37.4

Ponderosa Pine 28

White Pine 26

5. Solve for the missing information. (Make sure your units are correct) a) b) c) d)

V = _______________

D = _______________

m = _______________

D = _______________


6. Mike sold three board games on eBay to a single buyer. To save on shipping, he placed the three games into a single box with dimensions 12 cm x 12 cm x 10 cm. The box weighs 14.5 lbs.

a) What is the density of this box? (round to nearest hundredth)

b) The shipping cost was $42.40. What was the cost per pound to ship this package?

7. In August of 2015, a 16-year-old girl was diving in a German lake when she saw a gold bar glimmering at the bottom. The bar had its weight of 500 grams inscribed on it.

a) If gold’s density is 19.32 g/cm3 and the gold bar measured 6.5 cm by 4 cm, what was the height of the bar? (round to the nearest tenth centimeter?

b) If the current market value for gold is $40.90 per gram, what is the bar worth? __________________ 8. While wandering through a flea market in Dubai, Jenny sees three objects marked as gold but isn’t sure if they are telling the truth. The market seller listed the dimensions and the weight. Help Jenny determine which of the objects are likely to be gold.

Gold Ball Gold Column Gold Pyramid

a) The Gold Ball (10 g) (diameter 1 cm)

b) The Gold Column (90 g) (diameter 2 cm & height 10 cm)

c) The Gold Pyramid (77 g) (Base 2 cm by 2 cm, height 3 cm)

9. Jeff needs to move from Las Vegas, NV to Fargo, ND. He decides to rent a 20-foot shipping box. He calls two companies to compare prices. Ed’s Storage charges a flat rate for the box, $2500. Sally’s Super Storage rents the 20-foot container based on weight, $0.55 per pound (with a minimum weight of 1000).

a) If Jeff’s stuff weighs approximately 5,200 pounds, which would be the more affordable company? (show work)

b) What would the density of the container be?


10. A hemispherical water tank has a diameter of 20 inches. If water weighs 16.387 grams per cubic inch,

what is the approximate weight of the water when the tank is completely full, to the nearest gram? What is

that to the nearest pound if 1 lb = 453.5923 grams?

___________ g

___________ lbs

11. If the trunk of a Paper Birch tree (see question #4a) has a circumference of 4 feet and a height of 10 feet,

what is the approximate weight of the tree, to the nearest pound?

12. A wood carver forms a perfect sphere with a large piece of wood. The diameter of the sphere is 4 feet. She doesn’t remember which type of wood she was using, but she knows the sphere weighs 1,230 lbs. Use the chart in question in #4a to help you to determine the type of wood she probably was using.

13. Graham is building a retaining wall at the back of his property. He wants to use the concrete bricks (11.5 in. x 7 in. x 4 in., density 0.07 lb/in3) that his local outdoor gardening store sells. He needs 500 bricks but his trailer can only support 3,500 lb. How many trips will he need to make?

14. Wandering the streets of his city, Jim found three objects laying around and wanted to know their approximate value. Approximate what each item is worth.

Metal Density g/cm3 Cost per gram

Lead 11.3 $0.00146

Copper 8.6 $0.00540

Aluminum 2.7 $0.00132

Aluminum Lock Fake Lead Coin Copper Pipe Cap

5 cm x 4 cm x 1.5 cm 3 cm wide, 0.5 thick 2 cm diameter, 2 cm tall


1. What is population density?

2. Determine the population density of the following. a) b) c) d) e)

3. Rank the following states in the order that you think has the highest population density to the lowest.

Texas ______ Montana ______

New Jersey ______ California ______

4. Which is bigger: a square mile (mi2) or a square kilometer (km2)? (1 km = 0.621371 mile)

5. Convert the population denisity of 4.54 people/mi2 to people/km2. (1 mile = 1.60934 km)

6. Determine the population density in square miles for the following situations.

1 person per square

mile

a)

b)

c)

2

1 person

mi

d) e) f) g)

7. The American Bison population in Yellowstone National Park fluctuates from 2,300 to 5,500 animals in

two subpopulations, defined by where they gather for breeding. The northern herd breeds in the Lamar

Valley and on the high plateaus around it. The central herd breeds in Hayden Valley. Currently in

Yellowstone National park the bison count Is 4,750. If the park is 3471 square miles, what is the population

density of the American Bison per square mile?


8. Use these satellite images of neighborhoods in Las Vegas, Nevada to estimate their population density. a) The Canyons Custom Homes b) Monaco Estates Development c) Crystal Palace Apartments

(City Block 600 ft by 600 ft) (City Block 600 ft by 600 ft) (City Block 600 ft by 600 ft)

Single Dwelling Zoning Expensive custom homes

# of homes ______

Single Dwelling Zoning Track Home Development

# of homes ______

Multiple Dwelling Zoning Apartment Complex

Single Size (12 units each building) Double Size (24 units each building)

(2.58 people per household – US Average) (2.58 people per household – US Average) # of single size apartments ______

# of total people __________ # of total people __________ # of double size apartments _____

Square footage _____________ Convert to square miles….

Same square mileage as part a

# of total units ______

(1 ft = 0.000189394 mile)

(2.58 people per household – US Average)

# of people __________ Same square mileage as part a

Population Density ___________ (people per square mile)

Population Density ___________ (people per square mile)

Population Density ____________ (people per square mile)

9. In 2011, drought conditions brought swarms of grasshoppers to Texas. When a population of

grasshoppers is equal to or greater than 9.6 per square meter, it is classified as an infestation. One farmer

created a mesh cage that had a 6 ft by 6 ft opening. When he dropped it onto one of his crop fields, he

caught 36 grasshoppers. Was this truly an infestation? (1 foot = 0.3048 meters)


1. Use the map to obtain the approximate area of the state and then calculate the population density. Show work and calculations.

Oregon

Determine the area of

Oregon

Idaho


Idaho

If Oregon’s population is 4,028,977 --- determine its population density.

If Idaho’s population is 1,654,930 --- determine its population density.

Nevada


Nevada

California

Determine the area of California

If Nevada’s population is 2,890,845 --- determine its population density.

If California’s population is 39,144,818 --- determine its population density.


2. Calculate the area of the state and then estimate the population. Show your work and calculations. a) Connecticut

The population density: 2

741 people

mi

Connecticut Pop. ____________________

b) New York


420 people

mi

New York Pop. ____________________ c) New Jersey


1218 people

mi

New Jersey Pop. ____________________


3. To determine the plant population density many scientists use a quadrat sampling approach. A quadrat is a 1 meter by 1 meter square. The scientist defines a number of random locations at the site and then counts the number of plant species in the quadrat and records them. The calculation of plant population density (D) is the sum of all plants found (S) in quadrats divided by the number of locations (Q). In this study they have created 8 quadrats.

SD

Q

a) Complete the observation table.

Type 1 2 3 4 5 6 7 8 Total Plants

(S) Total # of Quadrats

(Q)

Plant Population Density

(plant/m2)

Blue Lily

b) Compare your quadrat sampling plant population density to the actual population for this entire region.

Are they close?

Documents

Geometry Workbook 13unionparish.enschool.org/ourpages/auto/2018/11/1/70881963/Workbook13.pdfNov 01, 2018 · I will be able to determine and calculate appropriate measures to real