A mathematical model is an equation that describes a real-world object or process. Modeling is the process of finding such equations. Once the model or equation has been found, it is then used to obtain information about the thing being modeled.

#### Example of Mixtures and Concentration

A manufacturer of soft drinks advertises their orange soda as “naturally flavored,” although it contains only 5% orange juice. A new federal regulation stipulates that to be called “natural,” a drink must contain at least 10% fruit juice. How much pure orange juice must this manufacturer add to 900 gal of orange soda to conform to the new regulation?

Solution Identify1 step : Identify the variable. Let x = the amount 1in gallons2 of pure orange juice to be added

2 step. Translate from words to algebra. In any problem of this type—in which two different substances are to be mixed—drawing a diagram helps us to organize the given information.

Set up the model. To set up the model, we use the fact that the total amount of orange juice in the mixture is equal to the orange juice in the first two vats.

45 + x =0.1(900 + x)

Solve it.

45 + x = 90 + 0.1x

0.9 x = 45

x=50

The manufacturer should add 50 gal of pure orange juice to the soda

#### Problems About the Time Needed to Do a Job

Time Needed to Do a Job

Because of an anticipated heavy rainstorm, the water level in a reservoir must be lowered by 1 ft. Opening spillway A lowers the level by this amount in 4 hours, whereas opening the smaller spillway B does the job in 6 hours. How long will it take to lower the water level by 1 ft if both spillways are opened?

SOLUTION Identify the variable. We are asked to find the time needed to lower the level by 1 ft if both spillways are open. So let

x = the time [in hours] it takes to lower the water level by 1 ft if both spillways are open

Translate from words to algebra. Finding an equation relating x to the other quantities in this problem is not easy. Certainly x is not simply 4 + 6, because that would mean that together the two spillways require longer to lower the water level than either spillway alone. Instead, we look at the fraction of the job that can be done in 1 hour by each spillway.

Set up the model. Now we set up the model

Solve. Now we solve for x

3x + 2x = 12

5x = 12

t will take 2 h 24 min, to lower the water level by 1 ft if both spillways are open.

#### Problems About Distance, Rate, and Time

The main formula is

A Distance-Speed-Time Problem:

A jet flew from New York to Los Angeles, a distance of 4200 km. The speed for the return trip was 100 km/h faster than the outbound speed. If the total trip took 13 hours of flying time, what was the jet’s speed from New York to Los Angeles?

Solution Identify the variable. We are asked for the speed of the jet from New York to Los Angeles. So let

s = speed from New York to Los Angeles

Then s + 100 = speed from Los Angeles to New York

Translate from words to algebra. Now we organize the information in a table. We fill in the “Distance” column first, since we know that the cities are 4200 km apart. Then we fill in the “Speed” column, since we have expressed both speeds (rates) in terms of the variable s. Finally, we calculate the entries for the “Time” column, using

Set up the model. The total trip took 13 hours, so we have the model

4200 / s + 4200 (s + 100) = 13

Solve it.

s = 600 or s=-53.8

Since s represents speed, we reject the negative answer and conclude that the jet’s speed from New York to Los Angeles was 600 km/h.

#### Energy Expended in Bird Flight

Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours, because air generally rises over land and falls over water in the daytime, so flying over water requires more energy. A bird is released from point A on an island, 5 mi from B, the nearest point on a straight shoreline. The bird flies to a point C on the shoreline and then flies along the shoreline to its nesting area D, as shown in Figure 5. Suppose the bird has 170 kcal of energy reserves. It uses 10 kcal/mi flying over land and 14 kcal/mi flying over water.

(a) Where should the point C be located so that the bird uses exactly 170 kcal of energy during its flight?

(b) Does the bird have enough energy reserves to fly directly from A to D?

Solution

(a) Identify the variable. We are asked to find the location of C. So let

x = distance from B to C

Translate from words to algebra. From the figure, and from the fact that

energy used = energy per mile x miles flown

we determine the following:

Set up the model. Now we set up the model.

Solve.