1.12 Applied Word Problems

  • Procedure for Solving Word Problems • Identifying the Unknown Quantities • Setting Up the Proper Equation • Examples of Solving Word Problems

Many applied problems are at first word problems, and we must put them into mathematical terms for solution. Usually, the most difficult part in solving a word problem is identifying the information needed for setting up the equation that leads to the solution. To do this, you must read the problem carefully to be sure that you understand all of the terms and expressions used. Following is an approach you should use.

Read the following examples very carefully and note just how the outlined procedure is followed.

EXAMPLE 1 Sum of forces on a beam

A 17-lb beam is supported at each end. The supporting force at one end is 3 lb more than at the other end. Find the forces.

Since the force at each end is required, we write

step 2

let F = the smaller force (in lb)

as a way of establishing the unknown for the equation. Any appropriate letter could be used, and we could have let it represent the larger force.

Also, since the other force is 3 lb more, we write

step 3

F + 3 = the larger force (in lb)

step 4 We now draw the sketch in Fig. 1.19.

A diagram of a 17 pound beam. Seventeen pounds of force pull down from the center. Force F pushes up on the far left, and force F plus 3 pushes up on the far right.

Fig. 1.19

Since the forces at each end of the beam support the weight of the beam, we have the equation

step 5

F + (F + 3) = 17

This equation can now be solved: 2F = 14

step 6

F = 7 lb

step 7 Thus, the smaller force is 7 lb, and the larger force is 10 lb. This checks with the original statement of the problem.

CAUTION

Always check a verbal problem with the original statement of the problem, not the first equation, because it was derived from the statement.

EXAMPLE 2 Office complex energy-efficient lighting

In designing an office complex, an architect planned to use 34 energy-efficient ceiling lights using a total of 1000 W. Two different types of lights, one using 25 W and the other using 40 W, were to be used. How many of each were planned?

Since we want to find the number of each type of light, we

let x = number of 25 W lights

Also, since there are 34 lights in all,

34 − x = number of 40 W lights

We also know that the total wattage of all lights is 1000 W. This means

A process for finding number of 25 watt and 40 watt lights used in an office complex is illustrated in 4 steps.

Therefore, there are 24 25-W lights and 10 40-W lights. The total wattage of these lights is 24(25) + 10(40) = 600 + 400 = 1000. We see that this checks with the statement of the problem.

EXAMPLE 3 Number of medical slides

A medical researcher finds that a given sample of an experimental drug can be divided into 4 more slides with 5 mg each than with 6 mg each. How many slides with 5 mg each can be made up?

We are asked to find the number of slides with 5 mg, and therefore we

let x = number of slides with 5 mg

Because the sample may be divided into 4 more slides with 5 mg each than of 6 mg each, we know that

x − 4 = number of slides with 6 mg

Because it is the same sample that is to be divided, the total mass of the drug on each set of slides is the same. This means

A process for finding number of slides with 5 milligram is illustrated in 3 steps.

Therefore, the sample can be divided into 24 slides with 5 mg each, or 20 slides with 6 mg each. Since the total mass, 120 mg, is the same for each set of slides, the solution checks with the statement of the problem.

EXAMPLE 4 Distance traveled—space travel

A space shuttle maneuvers so that it may “capture” an already orbiting satellite that is 6000 km ahead. If the satellite is moving at 27,000 km/h and the shuttle is moving at 29,500 km/h, how long will it take the shuttle to reach the satellite? (All digits shown are significant.)

First, we let t = the time for the shuttle to reach the satellite. Then, using the fact that the shuttle must go 6000 km farther in the same time, we draw the sketch in Fig. 1.20. Next, we use the formula distance = rate × time(d = rt) .  This leads to the following equation and solution.

A diagram of a shuttle and satellite that are 6,000 kilometers apart. The satellite follows an angled path to a given point, and the shuttle follows a direct path to meet it there.

Fig. 1.20

29,500 t = 6000 + 27,000 t. 2,500 t = 6,000. T = 2.400 h.

This means that it will take the shuttle 2.400 h to reach the satellite. In 2.400 h, the shuttle will travel 70,800 km, and the satellite will travel 64,800 km. We see that the solution checks with the statement of the problem.

EXAMPLE 5 Mixture—gasoline and methanol

A refinery has 7250 L of a gasoline-methanol blend that is 6.00% methanol. How much pure methanol must be added so that the resulting blend is 10.0% methanol?

First, let x = the number of liters of methanol to be added. The total volume of methanol in the final blend is the volume in the original blend plus that which is added. This total volume is to be 10.0% of the final blend. See Fig. 1.21.

A diagram of 3 containers with liters of methanol. A contain with 7,250 liters of 6.00% methanol is added to a contain with x liters of 100% methanol, equalling 7,250 plus x liters of 10.0% methanol.

Fig. 1.21

A process for finding the amount of pure methanol that must be added to produce a 10 percent methanol has 3 steps.

Checking (to three significant digits), there would be 757 L of methanol of a total 7570 L.

EXERCISES 1.12

In Exercises 1–4, make the given changes in the indicated examples of this section and then solve the resulting problems.

  1. In Example 2, in the first line, change 34 to 31.

  2. In Example 3, in the second line, change “4 more slides” to “3 more slides.”

  3. In Example 4, in the second line, change 27,000 km/h to 27,100 km/h.

  4. In Example 5, change “pure methanol” to “of a blend with 50.0% methanol.”

In Exercises 5–34, set up an appropriate equation and solve. Data are accurate to two significant digits unless greater accuracy is given.

  1. A certain new car costs $5000 more than the same model new car cost 6 years ago. Together a new model today and 6 years ago cost $64,000. What was the cost of each? (Assume all values are exact.)

  2. The flow of one stream into a lake is 1700 ft3 / s more than the flow of a second stream. In 1 h, 1.98 × 107ft3 flow into the lake from the two streams. What is the flow rate of each?

  3. Approximately 6.9 million wrecked cars are recycled in two consecutive years. There were 500,000 more recycled the second year than the first year. How many are recycled each year?

  4. A business website had twice as many hits on the first day of a promotion as on the second day. If the total number of hits for both days was 495,000, find the number of hits on each day.

  5. Petroleum rights to 140 acres of land are leased for $37,000. Part of the land leases for $200 per acre, and the reminder for $300 per acre. How much is leased at each price?

  6. A vial contains 2000 mg, which is to be used for two dosages. One patient is to be administered 660 mg more than another. How much should be administered to each?

  7. After installing a pollution control device, a car’s exhaust contained the same amount of pollutant after 5.0 h as it had in 3.0 h. Before the installation the exhaust contained 150 ppm/h (parts per million per hour) of the pollutant. By how much did the device reduce the emission?

  8. Three meshed spur gears have a total of 107 teeth. If the second gear has 13 more teeth than the first and the third has 15 more teeth than the second, how many teeth does each have?

  9. A cell phone subscriber paid x dollars per month for the first 6 months. He then increased his data plan, and his bill increased by $10 per month for the next 4 months. If he paid a total of $890 for the 10-month period, find the amount of his bill before and after the increase.

  10. A satellite television subscriber paid x dollars per month for the first year. Her monthly bill increased by $30 per month for the second and third years, and then another $20 for the fourth and fifth years. If the total amount paid for the 5-year period was $7320, find the three different monthly bill amounts.

  11. The sum of three electric currents that come together at a point in a circuit is zero. If the second current is twice the first and the third current is 9.2 μA more than the first, what are the currents? (The sign indicates the direction of flow.)

  12. A delivery firm uses one fleet of trucks on daily routes of 8 h. A second fleet, with five more trucks than the first, is used on daily routes of 6 h. Budget allotments allow for 198 h of daily delivery time. How many trucks are in each fleet?

  13. A natural gas pipeline feeds into three smaller pipelines, each of which is 2.6 km longer than the main pipeline. The total length of the four pipelines is 35.4 km. How long is each section?

  14. At 100% efficiency two generators would produce 750 MW of power. At efficiencies of 65% and 75%, they produce 530 MW. At 100% efficiency, what power would each produce?

  15. A wholesaler sells three types of GPS systems. A dealer orders twice as many economy systems at $40 each, and 75 more econoplus systems at $80 each, than deluxe systems at $140 each, for $42,000. How many of each were ordered?

  16. A person won a state lottery prize of $20,000, from which 25% was deducted for taxes. The remainder was invested, partly for a 40% gain, and the rest for a 10% loss. How much was each investment if there was a $2000 net investment gain?

  17. Train A is 520 ft long and traveling at 60.0 mi/h. Train B is 440 ft long and traveling at 40 mi/h in the opposite direction of train A on an adjacent track. How long does it take for the trains to completely pass each other? (Footnote: A law was once actually passed by the Wisconsin legislature that included “whenever two trains meet at an intersection … , neither shall proceed until the other has.”)

  18. A family has $3850 remaining of its monthly income after making the monthly mortgage payment, which is 23.0% of the monthly income. How much is the mortgage payment?

  19. A ski lift takes a skier up a slope at 50 m/min. The skier then skis down the slope at 150 m/min. If a round trip takes 24 min, how long is the slope?

  20. Before being put out of service, the supersonic jet Concorde made a trip averaging 120 mi/h less than the speed of sound for 1.0 h, and 410 mi/h more than the speed of sound for 3.0 h. If the trip covered 3990 mi, what is the speed of sound?

  21. Trains at each end of the 50.0-km-long Eurotunnel under the English Channel start at the same time into the tunnel. Find their speeds if the train from France travels 8.0 km/h faster than the train from England and they pass in 17.0 min. See Fig. 1.22.

    A diagram of two trains heading toward each other through a 50.0 kilometer tunnel. They meet after 17.0 minutes at point that is closer to England than France.

    Fig. 1.22

  22. An executive would arrive 10.0 min early for an appointment if traveling at 60.0 mi/h, or 5.0 min early if traveling at 45.0 mi/h. How much time is there until the appointment?

  23. One lap at the Indianapolis Speedway is 2.50 mi. In a race, a car stalls and then starts 30.0 s after a second car. The first car travels at 260 ft/s, and the second car travels at 240 ft/s. How long does it take the first car to overtake the second, and which car will be ahead after eight laps?

  24. A computer chip manufacturer produces two types of chips. In testing a total of 6100 chips of both types, 0.50% of one type and 0.80% of the other type were defective. If a total of 38 defective chips were found, how many of each type were tested?

  25. Two gasoline distributors, A and B, are 228 mi apart on Interstate 80. A charges $2.90/gal and B charges $2.70/gal. Each charges 0.2¢ / gal per mile for delivery. Where on Interstate 80 is the cost to the customer the same?

  26. An outboard engine uses a gasoline-oil fuel mixture in the ratio of 15 to 1. How much gasoline must be mixed with a gasoline-oil mixture, which is 75% gasoline, to make 8.0 L of the mixture for the outboard engine?

  27. A car’s radiator contains 12 L of antifreeze at a 25% concentration. How many liters must be drained and then replaced by pure antifreeze to bring the concentration to 50% (the manufacturer’s “safe” level)?

  28. How much sand must be added to 250 lb of a cement mixture that is 22% sand to have a mixture that is 25% sand?

  29. To pass a 20-m long semitrailer traveling at 70 km/h in 10 s, how fast must a 5.0-m long car go?

  30. An earthquake emits primary waves moving at 8.0 km/s and secondary waves moving at 5.0 km/s. How far from the epicenter of the earthquake is the seismic station if the two waves arrive at the station 2.0 min apart?

Answer to Practice Exercise

  1. 24 with 5 mg, 20 with 6 mg

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