Problems with Industrial & Health Applications
The authors of this sourcebook have attempted to collect problems from the area industries that we surveyed. Every attempt was made to use realistic numerical values, and most of the problems do reflect the types of mathematics encountered on the job.
A few of the problems were included to serve as a springboard for student discussion, and to sharpen the logical and mathematical skills students need in what teachers call the "real world."
It is intended that teachers, students, and interested business and industry leaders will send problems to the authors at this website and give us your comments to include in our storehouse. As this page builds, we may re-categorize the problems by type, by difficulty, or by discipline. We need your input.
The authors understand that the problems contained within this site serve only to begin the process of collecting good mathematical problems to connect the student to the "real world."
- A road in a subdivision is to have a slope of 1% (that is a one foot rise or fall for every 100 horizontal feet). How much fall does the road have over a distance of 840 feet? (Note: a 1% slope is considered the minimum for good drainage.)
- An Interstate highway cannot have a slope greater than 6%. If a highway with this slope goes down a mountain, how much elevation change would there be over a half mile horizontal distance?
- If a highway rises 84 feet during a horizontal travel of 1600 feet, what is the % slope? (Note: this slope is similar to the slope of I-24 at Monteagle mountain.)
Teaching point: Why would drivers, especially truck drivers, need to know about the slope of a highway?
- The county needs to gravel a new road 2 miles long and 16 feet wide. The gravel needs to be 6 inches deep to meet specifications. How many cubic yards of gravel does the county need?
- If there are 1.8 tons of gravel per cubic yard, how many tons of gravel will the county need for the job?
- A typical dump truck holds 25 tons of gravel. How many dump trucks will it take to deliver the gravel?
- If a good grade of gravel costs $8.50 per ton delivered, what will be the cost of the gravel for this road?
- A roll of printing stock paper weighs about 3000 pounds. How many rolls of this paper can be loaded onto a trailer if the maximum capacity of the trailer is 18 tons?
- A child is entered into the hospital after ingesting 12 aspirin tablets. The Merck Index indicates that renal failure can occur if as little as 3 grams is ingested, and may be fatal if as much as 10 grams is eaten. If each aspirin tablet contains 300 mg of aspirin, is the child in danger of death or renal failure?
- The dosage of a certain drug is 1.5 milligrams per kilogram of body weight. How much drug should be administered to a 140 pound woman?
- An IV order states that 1000 cc of D5W must be given over 8 hours. The IV tubing delivers 15 drops/cc. What drop rate (drops/min) must be administered?
- A patient receives an IV solution of D5W contains 5% dextrose dissolved in water. Dextrose is also called glucose, and delivers 4 kilocalories of energy per gram. If the density of this solution is 1.02 g/cc.and a patient is administered 1000 cc of D5W, how many kilocalories will the patient receive?
Teaching point: A kilocalorie is also called a big Calorie. These are nutrition Calories.
- Aspirin is dispensed using the old apothecary unit "grain". A standard tablet contains 5 grains of aspirin. The maximum effective dosage of aspirin is 650 milligrams. If 60 milligrams is equal to one grain, how many aspirin tablets should a person take?
- A patient loses 2 pints of blood in an accident. To prevent shock, determine how many cc's of IV solution must be given to replace this amount if whole blood is not available?
- In some states, like Alabama, airplanes are used to check speeds. To do this, a plane files over a highway marked with white lines every quarter mile. The pilot/trooper determines the time for a car to traverse from one white line to another. The resulting time is then used to calculate the speed of the car. The pilot/trooper then radios an awaiting patrol car, which then may surprise the unlucky motorist with a ticket. If the pilot/trooper determines that a car traveled a quarter mile in 15 seconds, what is the speed of the car?
- If the posted speed limit is 70 miles per hour, what is the shortest time that a car can legally travel a quarter mile?
- A car traveled a quarter mile in 18 seconds. If the posted speed limit is 55 mph, was the car speeding?
- The strength of the acid in a vat of cleaning solution is to be determined. During a titration, A 10 cc sample of the acid completely neutralizes 8 mL of a 2 M (molar) solution of sodium hydroxide. What is the concentration of the acid?
Teaching point: The formula V1 x C1 = V2 x C2 may be used to solve this problem. V is volume and C is concentration, the subscript "1" is used for initial values and the subscript "2" represents the final values. Molarity is a chemical concentration unit. This type of problem is found in quality control labs where acids are used to clean and prepare metals for coatings, and the strength of the acid is monitored closely.
- Sodium hypochlorite in a water solution is commonly known as Clorox or Purex. It has a strength of approximately 5% (5 grams dissolved in 100 mL of water solution.) It may be used for irrigating the bladder at a strength of 0.125%. How much 5% sodium hypochlorite solution should be used to make 500 mL of a 0.125% solution? Use the same formula as shown in the teaching point in the previous problem.
- A chemistry teacher has a laboratory with 16 students. For a certain lab, each student needs to perform 4 titrations. For each titration, each student will use at least 12 mL of 0.100 M hydrochloric acid. At least how much 0.100 M HCl should the teacher need for the lab?
- If the only acid the teacher has available is 12 M hydrochloric acid (the strongest available), then for the problem above, how much 12 M HCl should she use to prepare enough of the 0.100 M HCl for the students to use?
- A typical automobile brake pad is 1/2 inch thick. If an average person wears off 6.8 x 10-4 inches of the brake pad in a day, how many years should the brake pad last?
- If the brake pad in the previous problem has a 30,000 mile guarantee, and a driver averages 46 brake applications per day, how much of the brake pad is worn off during each stop? How much is worn off per mile?
- A typical new car tire has 5/8 inches of usable tread. If the tire has a 70,000 mile guarantee, and if the driving conditions remain constant, how much tread should be worn off per mile?
- A cheapskate determines that his old tires have 3/16 of an inch of tread left. If the cheapskate uses the treadwear per mile from the previous problem, how much longer before the tire is completely slick?
Best Buy Problems
- A manufacturer can purchase soap solution in two ways: 500 mL for $13.00 or 4 L for $53.57. Which product is the better buy?
- A quality control lab needs a constant supply of sodium hydroxide solution. The purchasing manager can either buy 20 liter bottles of 50% w/w sodium hydroxide solution for $181.81, or buy 50 kg drums pure sodium hydroxide beads for $476.68. Which is the better buy? The density of the 50% sodium hydroxide solution is 1.53 g/mL.
Estimation Problems for Fun
- Estimate the number of years an average person sleeps in a lifetime.
Teaching point: Get the students involved in these dimensional analysis problems by having them guess at how many hours an average person sleeps per night. They must consider that older people may only seep a few hours per night, and infants sleep many more hours per day. Do this in groups, and then compare the group results. Also, what is an average life span?
- Estimate the yearly aspirin usage by the residents of the state of Tennessee. How many 100 tablet bottles could be filled this be? If each bottle sells for an average of $4.75, calculate the amount money Tennesseans spend on aspirin.
Teaching point: What must the student know? Have them make an educated guess at their weekly use of aspirin, keeping in mind that older persons may use more.
Here is a problem we use in our chemistry classes to teach both dimensional analysis and density.
- A real stupid thief prepares to break into Fort Knox with an 5 gallon bucket and a blowtorch. His feeble plan is to melt down the gold bars with his blowtorch and pour the molten gold into the 5 gallon bucket. Then, in his foggy mind, he plans to carry the solidified bucket of gold into an awaiting van to make his getaway. Let us help this misguided person, and tell him how much the bucket of gold will weigh. The density of gold is 19.3 g/cc.
- If one roll of printing stock paper has a diameter of 4 feet and a height of 6 feet, calculate the volume of the roll in cubic feet and cubic meters.
- A manufacturer packs a car window motor into a box 12 cm long by 7 cm wide by 2 cm deep. The warehouse supervisor of Ford Motor Company needs to rent a warehouse to store 150,000 of these motors for the 1999 model year. How much space will he need to store these motors? Is a warehouse space 32 feet by 18 feet, with a height of 8 feet adequate for these motors?
- A company builds a warehouse 200 feet long by 90 feet wide. The ceiling is 18 feet high. The company needs two aisles 10 feet wide to run the length of the warehouse, and six aisles the same width to run the width of the warehouse. How much space is available for storage?
- The same company needs to store boxes 2 feet long by 18 inches wide by 2.5 feet deep in this warehouse. If the boxes can be stacked only 5 high, what is the maximum number of boxes that can be stored in the warehouse in the previous problem?
- A city office building is to have two floors, both rectangular: the first floor is to be 80 feet long, and 60 feet wide. If the second floor has the same dimensions, and the architect calculates the building to cost $36 per square foot, estimate the cost of the building.
- A spool holds 400 feet of steel wire that is 1/4 inch in diameter. According to reference books, the density of steel is approximately 7.6 g/cc. If the empty spool weighs 10 pounds, what is the weight of the spool and the wire?
Teaching point: The concept of density may need to be explained. Ask the students to calculate the density of steel in pounds/cubic inch, and pounds per cubic feet also. Many industries still use the English system, but most still use a mixture of English and Metric units.
- If finished steel wire costs approximately $40.00 per ton, how much does one spool of wire cost?
- The spool in the problems above unwinds and passes through a reducing die at a speed of 11 feet per minute. How long should it take for the spool to completely pass through the die?
- A wire with a diameter of 10 mm passes through a reducing die that lowers the wire diameter by a ratio of 4:1. What will be the final diameter of the wire?
The problem that follows was given to us by a worker at one of the factories we visited.
A tank 26 inches in diameter and 64 inches long is buried on its side. A hole is made on its topside, and a stick is to be inserted into the tank, which will contain gasoline. From noting the wet line on the stick, the amount of fuel in the tank can be determined. How should the stick be calibrated to read the gallons of fuel contained in the tank?
Teaching point: The problem may be worked with calculus using integration techniques. Also, a solution may be found using experimental techniques, such as pouring a gallon of fuel at a time into the tank and measuring how far it comes upon the stick. To start a classroom discussion , ask:
- Why not turn the tank on its end. Wouldn't the volume be easier to measure that way?
- Does it take the same effort to pump gasoline out of a tank turned on its side as it does if the tank is standing on its end?
- Is there a difference in fluid pressure on the seams of the tank if it is standing vs. lying on its side?
- Will the tank be more likely to leak straight up or on its side?
- How do gas stations bury gasoline storage tanks? Is there a reason they bury them the way they do in terms of cost?
- How do distilleries store their casks of fermenting liquor?
- What is the volume of the tank in the problem above in gallons? In liters?
- In some Latin countries, the cost of gasoline is 82 centavos per liter. If one cent is about 1.05 centavos (that is, the ratio of cent to centavos is 1: 1.05), calculate the cost of gasoline there in dollars per gallon.
Assume that you have just received a $30,000.00 grant to create a museum exhibit. You have 2000 square feet of temporary space in which to display it. Describe in detail, from conception to installation, the steps necessary to complete this project. You must not exceed your $30,000 and you must justify every expenditure.
- What kind of lighting will you use (e.g. track lighting, indirect lighting)?
- What kind of display cases will you use and who will build them?
- What about mounted exhibits on pedestals with perhaps acrylic hoods to protect them?
- What will be the storyline or theme of the exhibit?
- Who will decide the theme or storyline and develop it?
- What kind of signage will be used?
- What kind of art and graphics will be used? Who will do it?
- What problems will you encounter obtaining the desired objects to be displayed?
- What will be the cost?
- Can some objects be donated by individuals?
- What security measures will this involve?
- What can the community and the local businesses contribute?
- What phases of the project will absolutely have to be sub-contracted to someone?
- What else is there to think of?
- Remember that you must make maximum use of your $30,000.00.
Remember, we urge teachers and anyone working in industry to please send us any unique math application problems, and we will include them in our storehouse with credit given to the sender.