Selected Problems with Industrial and Health Applications
Last Updated 3/1/04
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."
Algebra
1. 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.)
2. 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?
3. 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?
4. 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?
5. If there are 1.8 tons of gravel per cubic yard, how many tons
of gravel will the county
need for the job?
6. A typical dump truck holds 25 tons of gravel. How many dump trucks
will it take to
deliver the gravel?
7. If a good grade of gravel costs $8.50 per ton delivered, what
will be the cost of the
gravel for this road?
8. 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?
9. 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?
10. 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?
11. 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?
12. 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.
13. 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?
14. 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?
15. 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?
16. If the posted speed limit is 70 miles per hour, what is the
shortest time that a car can
legally travel a quarter mile?
17. A car traveled a quarter mile in 18 seconds. If the posted speed
limit is 55 mph,
was the car speeding?
18. 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.
19. 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.
20. 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?
21. 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?
22. 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?
23. 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?
24. 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?
25. 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
26. 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?
27. 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
28. 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?
29. 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.
30. 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.
Geometry
1. 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.
2. 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?
3. 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?
4. 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?
5. 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.
6. 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.
7. If finished steel wire costs approximately $40.00
per ton, how much does one spool of
wire cost?
8. 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?
9. 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?
Calculus
The problem that follows was given to us by a worker at one of the factories we visited.
1. 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?
2. What is the volume of the tank in the problem above in gallons? In liters?
3. 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.
Research Problems
1. 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.
Teaching Points:
--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.