Deck 18: Growth and Form

Question 1

Other than the crushing strength considerations, what other factors limit the size of trees?

Answer 1

for example, root system

Question 2

You want to copy a poster whose dimensions are 24 inches by 30 inches onto a piece of paper 8.5 inches by 11 inches. You want the image to be as large as possible, but maintain the proportions of the original poster. What scaling factor will you use?

Answer 2

8.5/24, approximately 0.3542

Question 3

You want to copy a poster whose dimensions are 24 inches by 30 inches onto a piece of paper 8.5 inches by 11 inches. You want the image to be as large as possible, but maintain the proportions of the original poster. What are the dimensions of the image?

Answer 3

8.5 inches * 10.625 inches

Question 4

A miniature model for a sculpture weighs about 10 ounces and stands 8 inches high. The actual model is to be 3 feet high and weigh about 25 pounds. Can it be made from the same material as the model? If not, how should its weight per cubic inch compare with that of the model material?

Question 5

How does a whale's size affect the depth to which it can descend into the water?

Question 6

Are all circles similar? Why or why not?

Question 7

Create an example of an advertisement that misuses proportional change through its use of graphics.

Question 8

One of the famous problems of Greek antiquity was the duplication of the cube; that is, creating a cube similar to but double the volume of an existing cube. What if you instead wish to triplicate the cube, building a cube with three times the volume of an existing cube? What would be the scaling factor? Assuming the original cube was 1 cubic unit, what would be the dimensions of the larger cube?

Question 9

In 1997 a bottle of soda costs about 75 cents. Use the Consumer Price Index Table 21.5 to estimate when this soda would have cost 5 cents.

Question 10

Are all pentagons similar? Why or why not?

Question 11

Each year, the area's largest tomatoes and pumpkins are showcased at the county fair. If there is a maximum size for such produce, how can larger and larger specimens be grown each year?

Question 12

When comparing the relative dimensions of model trains you discover that their wheels are larger than they should be. Why might they be scaled incorrectly?

Question 13

Create an example of an advertisement which misuses proportional change through its use of percent growth.

Question 14

Are all rectangles similar? Why or why not?

Question 15

When comparing prices for pizzas, you discover that the large pizza has twice as much surface area of a medium pizza and costs less than two medium pizzas. However, your group decides to order two medium pizzas anyway. What additional factors could be omitted in the mathematical model?

Question 16

You want to copy a poster whose dimensions are 24 inches by 30 inches onto a piece of paper 11 inches by 17 inches. You want the image to be as large as possible, but maintain the proportions of the original poster. What scaling factor will you use?

Question 17

Why do spike heels cause more damage to hardwood floors than loafers?

Question 18

The cost of photographic paper is nearly proportional to the area of the paper. But some shops charge extra for "nonstandard" sizes. Why would they feel the need to do this?

Question 19

Use the Consumer Price Index Table 21.5 to estimate when a loaf of bread, 50 cents in 1970, doubled in price.

Question 20

You want to copy a poster whose dimensions are 24 inches by 30 inches onto a piece of paper 11 inches by 17 inches. You want the image to be as large as possible, but maintain the proportions of the original poster. What are the dimensions of the image?

Question 21

European car makers sometimes give fuel efficiency data in liters per 100 kilometers. A 2011 Fiat 500 gets 4.3 liters per 100 kilometers. What is this in miles per gallon?

Question 22

Large trucks often have multiple wheels on an axle. How would multiple wheels allow trucks to carry heavier loads?

Question 23

You own a painting whose dimensions are 24 inches by 30 inches. If you create a slide of this work so that the longest side of the image is 1 inch, what is the length of the smaller side of the slide image?

A) 1 inch
B) 4/5 inch
C) 16/25 inch
D) 1/5 inch

Question 24

Add the following measurements and give an answer with the correct number of significant digits: 1 mile, 1 foot, and 1 inch.

Question 25

On a recent visit to the gym, Clark ran 3 kilometers and Bruce ran 3 miles. Who ran the farthest? How much farther (in kilometers) did he run?

Question 26

A casino plans to redecorate its courtyard, and the plans include a large marble sculpture of a six-sided die. The sculpture will be 3 meters tall. The density of marble is 2700 kg/m³ and the crushing strength of marble is 7734 kg/m². Find the pressure on the bottom face of the sculpture. What problem does your answer pose with regard to the feasibility of the sculpture?

Question 27

A scale model of a new building is being built for display. A scale of 1 cm to 3 m is being used. It took 27,000 square centimeters of cardboard to construct the exposed surfaces of the model. How much wood (in square meters) will be required to construct the surfaces of the building?

Question 28

Multiply the following measurements and give an answer with the correct number of significant digits: 3.789 m, 2.000 m, and 7.1878 m.

Question 29

European car makers sometimes give fuel efficiency data in liters per 100 kilometers. A Ferrari 458 Italia gets 13 liters per 100 kilometers. What is this in miles per gallon?

Question 30

How many cubic centimeters are in 3 cubic meters?

Question 31

Why is allometric growth an appropriate model when the National Center for Missing and Exploited Children create sketches of missing people?

Question 32

Assume that the metabolic rate of mammals is proportional to surface area. If you scale up a cat into a supercat weighing 8 times as much, by what factor does its metabolic rate change?

Question 33

Two quantities x and y show a power curve relation y = 1.24x^3.1. Using base-10 logarithmic scales for both x and y axes, the curve is transformed to a line. Find the slope of the line.

Question 34

Hisashi has a radio-controlled plane that is a scale model of a Boeing 727. The model has a 2-foot wingspan, and the wingspan of a Boeing 727 is 108 feet. If the minimum flight speed for a Boeing 727 is about 165 miles per hour, what is the minimum flight speed for the model plane?

Question 35

You own a painting whose dimensions are 24 inches by 30 inches. If you create a slide of this work so that the longest side of the image is 1 inch, what is the scaling factor for the slide?

A) 1/24
B) 1/30
C) 1/720
D) 4/5

Question 36

You are planning to travel by a car from a city to another. The distance between two cities is 150 miles. Assuming that the efficiency of the car is 28 mpg, find the amount of gas in liters needed for the trip. Use 2 decimal places.

Question 37

Suppose that a quantity S is proportional to the surface area of an object. If another quantity M is proportional to the volume of the object, then for what power of M is the quantity S proportional to?

Question 38

Find the values of a and b so that the points (1.1, 14) and (1.4, 38) lie on the curve y = bx^a.

Question 39

Suppose that the efficiency of a car is measured as 34 mpg on highway. How many liters does it use for 100 km? Use 2 decimal places.

Question 40

Find the values of a and b so that the points (0.5, 21) and (1.4, 56) lie on the curve y = bx^a.

Question 41

In 1965 powdered gelatin cost 10 cents per box at the grocery store. If the 1965 CPI is 31.5 and the 1999 CPI is 166.2, compute this cost in 1999 dollars.

A) 13 cents
B) 19 cents
C) 48 cents
D) 53 cents

Question 42

In 1999 a compact car can cost approximately $14,000. What is this cost in 1990 dollars, the year that you bought your last car? (Assume the 1999 CPI is 166.2 and the 1990 CPI is 130.7.)

A) approximately $9300
B) approximately $13,940
C) approximately $11,010
D) approximately $17,803

Question 43

You bought a house for $130,000 in 1999. The previous owner paid $50,000 for the house in 1980. Comparing the converted cost, who paid more for the house? (Assume the 1999 CPI is 166.2 and the 1980 CPI is 82.4.)

A) You paid much more.
B) The previous owner paid much more.
C) You both paid approximately the same amount.

Question 44

Artificial Christmas trees tend to be proportional in shape. If a 4-foot tree needs about two strings of lights, what size tree would need five strings of lights?

A) a tree shorter than 6 feet
B) a tree between 6 and 8 feet tall
C) a tree between 8 and 9 feet tall
D) a tree taller than 9 feet

Question 45

You own a painting whose dimensions are 24 inches by 30 inches. If you create a slide of this work so that the longest side of the image is 1 inch, what is the area of the slide image?

A) 1 square inch
B) 4/5 square inch
C) 16/25 square inch
D) 5/4 square inches

Question 46

The cost of photographic paper is nearly proportional to the area of the paper. You own a painting whose dimensions are 24 inches by 30 inches. You create a photographic reproduction whose smaller side is 8 inches. How does its area compare with that of the original painting?

A) 0.3333
B) 0.2667
C) 0.1111
D) 0.0711

Question 47

Last summer you grew a giant squash that was 4.5 feet long and proportionally similar to your other squash. These squash tend to be about 1.5 feet long and weigh about 1 pound. About how much do you think the giant squash weighs?

A) less than 10 pounds
B) between 10 and 20 pounds
C) between 20 and 50 pounds
D) more than 50 pounds

Question 48

Christmas trees tend to be proportional in shape. You have a tree skirt that is circular and with diameter 3 feet, which you bought for a 7-foot tree. If you have a 5-foot tree this year, what size tree skirt will you need? (Round to the nearest size.)

A) a skirt whose diameter is 2.5 feet
B) a skirt whose diameter is 2 feet
C) a skirt whose diameter is 1.5 feet
D) a skirt whose diameter is 1 foot

Question 49

In 1976 a roast beef sandwich cost 89 cents at a local fast food shop. If the 1976 CPI is 56.9 and the 1999 CPI is 166.2, compute this cost in 1999 dollars.

A) 31 cents
B) $1.43
C) $1.93
D) $2.60

Question 50

The cost of photographic paper is nearly proportional to the area of the paper. You own a painting whose dimensions are 24 inches by 30 inches. You create a small photographic reproduction whose smaller side is 5 inches. What is the scaling factor for this reproduction?

A) 0.2083
B) 0.1667
C) 0.0434
D) 0.0278

Question 51

A model of a truck is built to a scale of 1 to 54. If the model will hold 7 cubic inches, how much will the actual truck hold?

A) 378 cubic feet
B) 638 cubic feet
C) 1544 cubic feet
D) 4536 cubic feet

Question 52

A model of a truck is built to a scale of 1 to 54. If the model casts a shadow of 8 square inches, how large a shadow will the actual truck cast?

A) 36 square feet
B) 162 square feet
C) 288 square feet
D) 729 square feet

Question 53

Artificial Christmas trees tend to be proportional in shape. If a 4-foot tree needs approximately 60 ornaments to cover the outer limbs of the tree, how many ornaments will a 6-foot tree need?

A) approximately 90 ornaments
B) approximately 120 ornaments
C) approximately 135 ornaments
D) approximately 150 ornaments

Question 54

Last summer you grew a giant squash that was 6 feet long and proportionally similar to your other squash. These squash tend to be about 1.5 feet long and weigh about 1 pound. About how much do you think the giant squash weighs?

A) less than 10 pounds
B) between 10 and 20 pounds
C) between 20 and 50 pounds
D) more than 50 pounds

Question 55

You own a painting whose dimensions are 24 inches by 30 inches. If you create a full-size copy of this painting as a poster, what is the scaling factor for the poster?

A) 1
B) 4/5
C) 1/24
D) 1/30

Question 56

The cost of photographic paper is nearly proportional to the area of the paper. You own a painting whose dimensions are 24 inches by 30 inches. You create a small photographic reproduction whose smaller side is 5 inches, and a large photographic reproduction whose smaller side is 8 inches. If the paper required for the small reproduction costs $1.50, how much will the paper required for the large reproduction cost?

A) $1.90
B) $2.40
C) $3.84
D) $4.50

Question 57

The minimum wage in 1999 was $5.15. What is the equivalent amount in 1942 dollars? (Assume the 1999 CPI is 166.2 and the 1942 CPI is 16.3.)

A) about 47 cents
B) about 49 cents
C) about 51 cents
D) about 53 cents

Question 58

A scale model of a truck is 1.5 inches high. If the actual truck is 9 feet high, what is the scaling factor of the model?

A) 1 to 6
B) 1 to 18
C) 1 to 36
D) 1 to 72

Question 59

You own a painting whose dimensions are 24 inches by 30 inches. If you create a postcard-sized reproduction of the painting whose area is 20 square inches, what is the scaling factor for the reproduction?

A) 1/4
B) 1/6
C) 1/36
D) 2/3

Question 60

A scale model of a truck is 5 inches long. If the model is built to a scale of 1 to 54, what is the length of the actual truck?

A) 270 feet
B) 101 feet
C) 36.7 feet
D) 22.5 feet

Question 61

The weight of a block of granite 1 ft × 2 ft × 4 ft is 1320 lb. If 12 blocks are used to build a wall 8 ft high, 12 ft long, and 1 ft wide, what is the pressure on the bottom faces?

A) 4.58 lb/in²
B) 9.17 lb/in²
C) 18.3 lb/in²
D) 36.7 lb/in²

Question 62

A granite sculpture weighs approximately 4500 lb. If a smaller version 2/3 the height of the original is created from granite, how much will it weigh?

A) approximately 3675 lb
B) approximately 3000 lb
C) approximately 2000 lb
D) approximately 1300 lb

Question 63

Trout costs about $6.50 per pound in the United States. If a Canadian dollar exchanges for 75 U.S. cents and a kilogram is about 2.2 pounds, what is the equivalent cost in Canadian dollars?

A) $2.22 per kilogram
B) $3.94 per kilogram
C) $10.73 per kilogram
D) $19.07 per kilogram

Question 64

A wooden carving of an eagle stands 2 inches high and weighs approximately 3 ounces. If a larger carving made from the same material stands 5 feet high, how much will it weigh?

A) approximately 50 lb
B) approximately 70 lb
C) approximately 170 lb
D) approximately 5000 lb

Question 65

A small steel spoon is 6 inches long and weighs approximately 4 ounces. If it is used as a model for a large steel spoon to be constructed from melting approximately 200 lb of steel spoons, how long will the large spoon be?

A) approximately 4.6 ft
B) approximately 7.4 ft
C) approximately 10 ft
D) approximately 14 ft

Question 66

Apples cost about $1.75 Canadian dollars per kilogram in Canada. If a Canadian dollar exchanges for 75 U.S. cents and a kilogram is about 2.2 pounds, what is the equivalent cost in U.S. dollars?

A) $0.60 U.S. per pound
B) $1.06 U.S. per pound
C) $2.89 U.S. per pound
D) $5.13 U.S. per pound

Question 67

The weight of a block of granite 1 ft × 2 ft × 4 ft is 1320 lb. If it stands on a large face (4 ft × 2 ft), what is the pressure on the bottom face?

A) 1.15 lb/in²
B) 2.29 lb/in²
C) 4.58 lb/in²
D) 9.17 lb/in²

Question 68

A granite sculpture weighs approximately 4500 lb. If a smaller version 3/4 the height of the original is created from granite, how much will it weigh?

A) approximately 3900 lb
B) approximately 3475 lb
C) approximately 2500 lb
D) approximately 1900 lb

Question 69

A spherical balloon has a circumference of 10 inches. If it expands to twice the volume, what happens to its surface area?

A) Its surface area more than doubles.
B) Its surface area doubles.
C) Its surface area increases but does not double.

Question 70

You bought a house in 1999 for $140,000. The previous owner paid $55,000 for the same house in 1976. Comparing the converted costs, who paid more for the house? (Assume the 1999 CPI is 166.2 and the 1976 CPI is 56.9.)

A) You paid more.
B) The previous owner paid more.
C) You both paid about the same.

Question 71

A sparrow has a minimum speed of about 20 miles per hour. If a robin is twice as long as a sparrow and proportional in shape, what is the minimum speed of a robin?

A) 40 miles per hour
B) 28 miles per hour
C) 14 miles per hour
D) 10 miles per hour

Question 72

A small steel spoon is 6 inches long and weighs approximately 4 ounces. If it is used as a model for a large steel spoon 4 feet long to be installed at the local cafe, how much would it weigh?

A) approximately 16 lb
B) approximately 32 lb
C) approximately 128 lb
D) approximately 2048 lb

Question 73

An artist wishes to melt 2000 copper pennies to create a large penny proportional to an ordinary penny. The diameter of a penny is approximately 3/4 in. What will be the diameter of the large penny?

A) approximately 0.78 ft
B) approximately 2.2 ft
C) approximately 5 ft
D) approximately 9.4 ft

Question 74

A spherical balloon has a circumference of 10 inches. If it expands to twice the volume, what happens to its circumference?

A) Its circumference more than doubles.
B) Its circumference doubles.
C) Its circumference increases but does not double.

Question 75

An artist wishes to melt copper pennies to create a large penny proportional to an ordinary penny and with diameter 5 inches. An ordinary penny has diameter 3/4 in. How many pennies will be needed?

A) approximately 10
B) approximately 50
C) approximately 150
D) approximately 300

Question 76

A sparrow has a minimum speed of about 20 miles per hour. If a blue jay is five times as long as a sparrow and proportional in shape, what is the minimum speed of a blue jay?

A) 100 miles per hour
B) 45 miles per hour
C) 9 miles per hour
D) 4 miles per hour

Question 77

The weight of a block of granite 1 ft × 2 ft × 4 ft is 1320 lb. If it stands on a small face (1 ft × 2 ft), what is the pressure on the bottom face?

A) 1.15 lb/in²
B) 2.29 lb/in²
C) 4.58 lb/in²
D) 9.17 lb/in²

Question 78

Assuming that the sail of the Dimetrodon developed to dissipate heat when an individual doubled in weight, how much did the sail grow?

A) It more than doubled in surface area.
B) It doubled in surface area.
C) It grew but did not double in surface area.

Question 79

Ground steak costs about $2.50 per pound in the United States. If an Australian dollar exchanges for 80 U.S. cents and a kilogram is about 2.2 pounds, what is the equivalent cost in Australian dollars?

A) $0.91 per kilogram
B) $1.42 per kilogram
C) $4.40 per kilogram
D) $6.88 per kilogram

Question 80

A spherical balloon has a circumference of 10 inches. If it expands so that its surface area is doubled, what happens to its volume?

A) Its volume more than doubles.
B) Its volume doubles.
C) Its volume increases but does not double.