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Section 8.7 Projects for Chapter 8

Projects 1–4 deal with the mean and median.

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Project 8.2. Ages of US women.

According to the Bureau of the Census, the average age of the U.S. population is steadily rising. The table gives data for two types of average, the mean and the median, for the ages of women. (Consult the Glossary for the definitions of mean and median.)

Date |
July 1990 |
July 1992 |
July 1994 |
July 1996 |
July 1998 |

Median age |
\(34.0\) |
\(34.6\) |
\(35.2\) |
\(35.8\) |
\(36.4\) |

Mean age |
\(36.6\) |
\(36.8\) |
\(37.0\) |
\(37.3\) |
\(37.5\) |

Which is growing more rapidly, the mean age or the median age?

Plot the data for median age versus the date, using July 1990 as \(t=0\text{.}\) Draw a line through the data.

What is the meaning of the slope of the line in part (b)?

Plot the data for mean age versus date on the same axes. Draw a line that fits the data.

Use your graph to estimate when the mean age and the median age will be the same.

For the data given, the mean age of women in the United States is greater than the median age. What does this tell you about the U.S. population of women?

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Project 8.3. Ages of US men.

Repeat

Project 8.2 for the mean and median ages of U.S. men, given in the table below.

Date |
July 1990 |
July 1992 |
July 1994 |
July 1996 |
July 1998 |

Median age |
\(31.6\) |
\(32.2\) |
\(32.9\) |
\(33.5\) |
\(34.0\) |

Mean age |
\(33.3\) |
\(34.0\) |
\(34.3\) |
\(34.5\) |
\(34.9\) |

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Project 8.4. Ages of US women Part II.

Find an equation for the line you graphed in part (b), median age versus date.

Find an equation for the line you graphed in part (d), mean age versus date.

Solve the system of equations algebraically. How does your answer compare to the solution you found by graphing?

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Project 8.5. Ages of US men Part II.

Find an equation for the line you graphed in part (b), median age versus date.

Find an equation for the line you graphed in part (d), mean age versus date.

Solve the system of equations algebraically. How does your answer compare to the solution you found by graphing?

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Project 8.6. Salinity and freezing point.

Water slowly contracts as it cools, causing it to become denser, until just before freezing, when the density decreases slightly. The salinity of water affects its freezing point and also the temperature at which it reaches its maximum density: The higher the salinity, the lower the temperature at which the maximum density occurs. Pure water is densest at \(4\degree\)C and freezes at \(0\degree\)C. Water that is \(15\%\) saline is densest at \(0.8\degree\)C and freezes at \(-0.8\degree\)C.

Write an equation for the water’s temperature of maximum density in terms of its salinity in percent.

Write an equation for the water’s freezing point in terms of its salinity in percent.

Graph the system of equations.

Solve the system to find the salinity of water that freezes when it is densest. What is the freezing point?

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Project 8.7. Charles’s Law.

Charles’s law says that the temperature of a gas is related to its volume by a linear equation. A gas that occupied one liter at \(0\degree\)C expanded to \(1.2\) liters at a temperature of \(54.6\degree\)C. A second sample of gas that occupied \(2\) liters at \(0\degree\)C expanded to \(2.8\) liters at \(109.2\degree\)C.

Find a linear equation for temperature, \(T\text{,}\) in terms of volume, \(V\text{,}\) for the first sample of gas, and a second linear equation for the second sample of gas.

Graph the system of equations.

Solve the system. What is the significance of the intersection point?

Show that both equations from part (a) can be written in the form \(T = c\left(\dfrac{V}{V_0}-1 \right)\text{,}\) where \(V_0\) is the volume of the gas at \(0\degree\)C.

Grazing animals spend most of their time foraging for food. A small animal such as a ground squirrel must also be alert for predators.Which foraging strategy favors survival: Should the animal look for foods that satisfy its dietary requirements in minimum time, thus minimizing its exposure to predators and the elements, or should it try to maximize its intake of nutrients?

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Project 8.8. Ground Squirrels.

The Columbian ground squirrel eats forb, a type of flowering weed, and grass. One gram of forb provides \(2.44\) kilocalories, and one gram of grass provides \(2.26\) kilocalories. For survival, the squirrel needs at least \(100\) kilocalories per day. However, the squirrel can digest no more than \(314\) wet grams of food daily. Each dry gram of forb becomes \(2.67\) wet grams in the squirrel’s stomach, and each dry gram of grass becomes \(1.64\) wet grams. In addition, the squirrel has at most \(342\) minutes available for grazing each day. It takes the squirrel \(2.05\) minutes to eat one gram of forb and \(5.21\) minutes to eat one gram of grass.

Write a system of inequalities for the number of grams of grass and forb the squirrel can eat and survive.

Graph the system.

Give two possible combinations of grass and forb that satisfy the squirrel’s feeding requirements.

If the squirrel can find only grass on a given day, how much grass will it need to eat? If the squirrel can find only forb, how much forb will it need?

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Project 8.9. Isle Royale Moose.

The Isle Royale moose eats deciduous leaves, which provide \(3.01\) kilocalories of energy per gram, and aquatic plants, which provide \(3.82\) kilocalories per gram. The moose needs at least \(10,946\) kilocalories of food per day, and its daily sodium requirement must be met by consuming at least \(453\) grams of aquatic plants. It takes the moose \(0.05\) minute to eat one gram of aquatic plants and \(0.06\) minute to eat each gram of leaves. To maintain its thermal balance, the moose cannot spend more than \(150\) minutes each day standing in water and eating aquatics plants, or more than \(256\) minutes eating leaves. Each gram of aquatic plants becomes \(20\) wet grams in the moose’s stomach, and each gram of leaves becomes \(4\) wet grams, and the stomach can hold no more than \(32,900\) wet grams of food daily.

Write a system of inequalities for the number of grams of aquatic plants and of leaves the moose can eat.

Graph the system.

Give two possible combinations of aquatic plants and leaves that satisfy the moose’s feeding requirements.

Can the moose survive by eating only aquatic plants?

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Project 8.10. Ground Squirrels Part II.

Project 8.8 described the diet of the Colombian ground squirrel. Refer to the system of constraints and the set of feasible solutions you found in that problem.

Write a formula for the squirrel’s energy consumption. What is the optimum diet for the squirrel if it wants to maximize its energy consumption? Round your answers to one decimal place.

Write a formula for the squirrel’s foraging time. What is the optimum diet for the squirrel if it wants to minimize its time foraging? Round your answer to one decimal place.

The actual observed diet of the squirrel consists of approximately \(100\) grams of forb and \(25\) grams of grass. Does the squirrel appear to be trying to maximize its caloric intake or to minimize its time eating? (Because \(70\%\) of the food available to the squirrel is grass, its diet is not determined by convenience.)

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Project 8.11. Isle Royale Moose Part II.

Project 8.9 described the diet of the Isle Royale moose. Refer to the system of constraints and the set of feasible solutions you found in that problem.

Write a formula for the moose’s energy consumption. What is the optimum diet for the moose if it wants to maximize its energy consumption? Round your answers to one decimal place.

Write a formula for the moose’s foraging time. What is the optimum diet for the moose if it wants to minimize its time foraging? Round your answer to one decimal place.

The actual observed diet of the moose consists of approximately \(868\) grams of aquatic plants and \(3437\) grams of leaves. Does the moose appear to be trying to maximize its caloric intake or to minimize its time eating?