Chapter 1

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The difference between 500 and the number of people in a theater gives the number of unsold tickets.

Cycling. A cyclist leaves his training base for a morning workout, riding at the rate of \(18 \mathrm{mph}\). One hour later, his support staff leaves the base in a car going 45 mph in the same direction. How long will it take the support staff to catch up with the cyclist?

Solve each formula for the specified variable. See Example 5. $$ V=\frac{1}{3} \pi r^{2} h \quad \text { for } h $$

List the elements of $$ \left\\{-3,-\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75,916 . \overline{6}\right\\} $$ that belong to the following sets. Prime numbers

Perform the operations. See Examples 4 and 5 . $$ \frac{-10.8}{-1.2} $$

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. Each test score was increased by 15 points to give a new adjusted test score.

Two marathon runners leave the starting gate, one running 12 mph and the other 10 mph. If they maintain the pace, how long will it take for them to be one- quarter of a mile apart?

Height of a Triangle. If the height of a triangle with a base of 8 inches is tripled, its area is increased by 96 square inches. Find the height of the triangle.

Perform the operations. See Examples 4 and 5 . $$ \frac{-13.5}{-1.5} $$

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The weight of a super-size order of French fries is twice that of a regular- size order.

Briefly explain what should be accomplished in each of the steps (analyze, assign, form, solve, state, and check) of the problem-solving strategy used in this section.

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The product of the number of boxes of crayons in a case and 12 gives the number of crayons in a case.

List the elements of $$ \left\\{-3,-\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75,916 . \overline{6}\right\\} $$ that belong to the following sets. Odd prime numbers

Translate each verbal model into a mathematical model. Answers may vary depending on the variables chosen. The perimeter of an equilateral triangle can be found by tripling the length of one of its sides.

At 2 P.M., two military convoys leave Eagle River, Wisconsin, one headed north and one headed south. The convoy headed north averages \(50 \mathrm{mph}\), and the convoy headed south averages 40 mph. They will lose radio contact when the distance between them is more than 35 miles. When will this occur?

Solve each equation. Check each result. See Example 5. $$ 3(k-4)=-36 $$

When expressed as a decimal, is \(\frac{7}{9}\) a terminating or repeating decimal?

Graph each set on a number line. $$ \left\\{-\frac{5}{2},-0.1,2.142765 \ldots, \frac{\pi}{3},-\sqrt{11}, 2 \sqrt{3}\right\\} $$

Write a mathematical model for each situation. Answers may vary depending on the variables chosen. Taxes. \(\quad\) A married couple has decided to split the money equally when they receive their federal income tax refund. Furthermore, the husband is going to donate \(\$ 75\) of his share to charity. Describe the relationship between the amount of money that the husband will keep and the amount of the couple's refund.

For her workout, Sarah walks north at the rate of 3 mph and returns at the rate of 4 mph. How many miles does she walk if the round trip takes 3.5 hours?

$$ \text { Solve: } x+20=4 x-1+2 x $$

Solve each formula for the specified variable. See Example 5. $$ H=17-\frac{A}{2} \text { for } A $$

Evaluate each expression. See Example \(6 .\) $$ 2^{5} $$

Graph each set on a number line. $$ \left\\{2 \frac{1}{9},-3.821134 \ldots,-\frac{\pi}{2}, \sqrt{15},-0.9, \frac{\sqrt{2}}{2}\right\\} $$

Write a mathematical model for each situation. Answers may vary depending on the variables chosen. Copiers. \(\quad\) A business is going to rent a copy machine. Under the rental agreement, the company is charged \(\$ 105\) per month and 3 \(\boldsymbol{\alpha}\) for every copy that is made. Describe the relationship between the monthly copier expense and the number of copies made.

How many pounds of red licorice bits that sell for \(\$ 1.90\) per pound should be mixed with 5 pounds of lemon gumdrops that sell for \(\$ 2.20\) per pound to make a candy Mixture that could be sold for \(\$ 2\) per pound?

Multiply. See Example 3 . $$\frac{2}{3}\left(3 s^{2}-9\right)$$

Solve each equation. Check each result. See Example 5. $$ 2(a-5)-(3 a+1)=0 $$

Evaluate each expression. See Example \(6 .\) $$ (-7.9)^{2} $$

Graph each set on a number line. $$ \left\\{3 . \overline{15}, \frac{22}{7}, 3 \frac{1}{8}, \pi, \sqrt{10}, 3.1\right\\} $$

Write a mathematical model for each situation. Answers may vary depending on the variables chosen. Bottled Water. \(\quad\) A driver left a production plant with 300 fivegallon bottles of drinking water on his truck. His delivery route consisted of office buildings, each of which was to receive 6 bottles of water. Describe the relationship between the number of bottles of water left on his truck and the number of stops that he has made.

A store sells regular green tea for \(\$ 16\) a pound and an exotic loose leaf tea for \(\$ 28\) a pound. To get rid of 40 pounds of the exotic loose leaf tea that are not selling, a shopkeeper makes a blend to put on sale for \(\$ 20\) a pound. How many pounds of green tea should he use?

Solve \(T-R=m a\) for \(R\)

Solve each formula for the specified variable. See Example 5. $$ \bar{v}=\frac{1}{2}\left(v+v_{0}\right) \quad \text { for } v_{0} $$

Evaluate each expression. See Example \(6 .\) $$ (-4.6)^{2} $$

Graph each set on a number line. $$ \left\\{-0 . \overline{331},-0.331,-\frac{1}{3},-\sqrt{0.11}\right\\} $$

Write a mathematical model for each situation. Answers may vary depending on the variables chosen. Collectibles. A woman inherited 9 antique dolls. She decided to add to her collection by purchasing two more dolls each month. Describe the relationship between the number of antique dolls in her collection and the number of months since she began to purchase them.

A pound of dried pineapple bits sells for \(\$ 6.19\), a pound of dried banana chips sells for \(\$ 4.19,\) and a pound of raisins sells for \(\$ 2.39\) a pound. Two pounds of raisins are to be mixed with equal amounts of pineapple and banana to create a trail mix that will sell for \(\$ 4.19\) a pound. How many pounds of pineapple and banana chips should be used?

Solve each equation. Check each result. See Example 5. $$ 9(x-2)=-6(4-x)+18 $$

Use the following property of levers: \(A\) lever will be in balance when the sum of the products of the forces on one side of a fulcrum and their respective distances from the fulcrum is equal to the sum of the products of the forces on the other side of the fulcrum and their respective distances from the fulcrum. Moving a Stone. A woman uses a 10 -foot bar to lift a 210 -pound stone. If she places another rock 3 feet from the stone to act as the fulcrum, how much force must she exert to move the stone?

The set of prime numbers less than 8

Use the following property of levers: \(A\) lever will be in balance when the sum of the products of the forces on one side of a fulcrum and their respective distances from the fulcrum is equal to the sum of the products of the forces on the other side of the fulcrum and their respective distances from the fulcrum. Lifting a Car. A 350 -pound football player brags that he can lift a \(2,500\) -pound car. If he uses a 12 -foot bar with the fulcrum placed 3 feet from the car, will he be able to lift the car?

Solve each formula for the specified variable. See Example 5. $$ P=2(l+w) \quad \text { for } l $$

Evaluate each expression. See Example \(6 .\) $$ -8^{2} $$

The set of integers between \(-7\) and 0

A wholesaler of premium organic planting mix notices that the retail garden centers are not buying her product because of its high price of \(\$ 1.57\) per cubic foot. She decides to mix sawdust with the planting mix to lower the price per cubic foot. If the wholesaler can buy the sawdust for \(\$ 0.10\) per cubic foot, how many cubic feet of each must be mixed to have \(6,000\) cubic feet of planting mix that could be sold to retailers for \(\$ 1.08\) per cubic foot?

Evaluate each expression. See Example \(6 .\) $$ \left(-\frac{3}{5}\right)^{3} $$

The set of odd integers between 10 and 18

A pound of tin is worth \(\$ 1\) more than a pound of copper. Four pounds of tin are mixed with 6 pounds of copper to make bronze that sells for \(\$ 3.65\) per pound. How much is a pound of tin worth?

Use the following property of levers: \(A\) lever will be in balance when the sum of the products of the forces on one side of a fulcrum and their respective distances from the fulcrum is equal to the sum of the products of the forces on the other side of the fulcrum and their respective distances from the fulcrum. Balancing a Seesaw. Jim and Bob sit at opposite ends of an 18-foot seesaw, with the fulcrum at its center. Jim weighs 160 pounds, and Bob weighs 200 pounds. Kim sits 4 feet in front of Jim, and the seesaw balances. How much does Kim weigh?