### Chapter 4 Review Exercises

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• 1. Apportionment

Apportionment is the proportional distribution of the 435 members of the U.S. House of Representatives based on the population of each state. In 1960, each member of the House represented about 415,000 people. On average, this number has increased by 11.5% each decade.

Write a formula that can be used to model the exponential growth of the population per representative every decade since 1960. (Note: n represents decades.)

• Worked-Out Solution

The formula for this exponential growth is

P = 415,000 people,  r = 11.5%,  n is the number of decades

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```  ______    ______   _    _     _____    __   __
/_   _//  /_   _// | \  / ||  |  ___||  \ \\/ //
`-| |,-    -| ||-  |  \/  ||  | ||__     \   //
| ||     _| ||_  | .  . ||  | ||__     / . \\
|_||    /_____// |_|\/|_||  |_____||  /_//\_\\
`-`'    `-----`  `-`  `-`   `-----`   `-`  --`

```
• 2. Making a Table to Show Estimated Population

Apportionment is the proportional distribution of the 435 members of the U.S. House of Representatives based on the population of each state. In 1960, each member of the House represented about 415,000 people. On average, this number has increased by 11.5% each decade. Make a table showing the estimated population per representative every decade from 1960 through 2010.

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```  _  __     ___      _  _    __   __    _____
| |/ //   / _ \\   | \| ||  \ \\/ //  /  ___||
| ' //   / //\ \\  |  ' ||   \ ` //  | // __
| . \\  |  ___  || | .  ||    | ||   | \\_\ ||
|_|\_\\ |_||  |_|| |_|\_||    |_||    \____//
`-` --` `-`   `-`  `-` -`     `-`'     `---`

```
• 3. Projecting Populations

Apportionment is the proportional distribution of the 435 members of the U.S. House of Representatives based on the population of each state. In 1960, each member of the House represented about 415,000 people. On average, this number has increased by 11.5% each decade.

Use the formula from Exercise 1 to project the population per representative every decade from 2020 through 2100.

• Worked-Out Solution

The formula for this exponential growth is

P = 415,000 people, r = 11.5%, n is the number of decades

A spreadsheet using this formula is shown below.

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```           __   __    _  _    __   __  __    __
___    \ \\/ //  | \| ||  \ \\/ // \ \\ / //
/   ||   \ ` //   |  ' ||   \ ` //   \ \/ //
| [] ||    | ||    | .  ||    | ||     \  //
\__ ||    |_||    |_|\_||    |_||      \//
-|_||    `-`'    `-` -`     `-`'       `
`-`
```
• 4. Graphing Populations

Apportionment is the proportional distribution of the 435 members of the U.S. House of Representatives based on the population of each state. In 1960, each member of the House represented about 415,000 people. On average, this number has increased by 11.5% each decade. Graph the estimated and projected populations from Exercises 2 and 3.

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``` __   _    _    _    ______      ___       _____
| || | || | || | || |      \\   / _ \\    / ___//
| '--' || | || | || |  --  //  | / \ ||   \___ \\
| .--. || | \\_/ || |  --  \\  | \_/ ||   /    //
|_|| |_||  \____//  |______//   \___//   /____//
`-`  `-`    `---`   `------`    `---`   `-----`

```
• 5. Comparing Populations

Apportionment is the proportional distribution of the 435 members of the U.S. House of Representatives based the population of each state. In 1960, each member of the House represented about 415,000 people. On average, this number has increased by 11.5% each decade.

Suppose the actual populations per representative for 2050 and 2060 are 1,100,000 and 1,230,000, respectively. What is the rate of growth per decade? How do these numbers compare to your model?

• Worked-Out Solution

If the actual population per representative increased from 1,100,000 to1,230,000 from 2050 to 2060, the rate of growth for this decade would be

A spreadsheet using the formula from Exercise 1 is shown below. The actual populations per representative for 2050 and 2060 closely follow the values in the model.

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``` ______    _    _     _  __    _____     ____
|      \\ | || | ||  | |/ //  |  ___||  |  _ \\
|  --  // | || | ||  | ' //   | ||__    | |_| ||
|  --  \\ | \\_/ ||  | . \\   | ||__    | .  //
|______//  \____//   |_|\_\\  |_____||  |_|\_\\
`------`    `---`    `-` --`  `-----`   `-` --`

```
• 6. Estimating Population

Apportionment is the proportional distribution of the 435 members of the U.S. House of Representatives based on the population of each state. In 1960, each member of the House represented about 415,000 people. On average, this number has increased by 11.5% each decade. Use the map to estimate the population of California.

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``` ______    __   __   _    _      ___     _    _