Real World Mathematics

"What is this good for?" Most math teachers have heard this question repeated with each new group of students. The following are a collection of mathematics applications on a course level


Developmental Mathematics

A recipe calls for specific quantities of ingredients, but is designed to feed ten people. If only seven people are coming for dinner, how do the ingredient quantities need to be altered?

On a canoeing trip upstream in water that flows at 4 miles per hour, it took you 90 minutes to travel 3 hours. How much time should you allow for the return trip?

You plan to tile your kitchen floor with prefabricated ceramic tiles that are 14 inches square. If your kitchen is 15 feet by 19 feet, how many tiles do you need to purchase?

Your school decides to go from 50-minute class meetings to 55-minute class meetings. At present, 45 are meetings are required to meet the state minimum number of in-class instructional hours. Under the new 55-minute standard, what is the minimum number of required class meetings?

Your new lawn tractor requires 6 gallons of a fuel mix that is 97% gasoline and 3% oil. You have access to a mix that is 95% gasoline and 5% oil. You also have pure oil. How much of each liquid should be mixed to produce a 97-3 fuel?

College Algebra

In a marketing class, you have chapter test scores of 73, 88, and 84. You make a 91 on the final exam and a 94 on your term paper. If the instructor informs you that the final counts twice as much as the other exams, but the term paper counts only half as much as a chapter exam, what is your grade in the course?

You are in the probationary period on a new job, and have not yet qualified for sick leave, but you miss one day due to a virus. If overtime in your workplace pays one and three-quarters of the regular rate, how many overtime hours must you work to make up for the lost wages from your sick day?

You and a friend are overseeing a Boy Scout hike. You are camped approximately 4 miles east of your rendezvous point, and the other group is 2 miles south. You are hiking over flat ground and can travel at almost 3 miles per hour. The other group is hiking uphill, and can only expect to move at a 1.2 mile per hour pace. Both groups have walkie-talkies with a range of 2 miles. If you both leave at 9:00, at what time can you expect to first make walkie-talkie contact with the other group?

You are a facilities manager for a small town. The town contains approximately 400 miles of road that must be plowed following a significant snowfall. How many plows must be used in order to complete the job in one day if the plows can travel at approximately 7 miles per hour when engaged?

Trigonometry/Precalculus

The chain assembly of a bicycle has a front sprocket with a radius of 5.5 inches and a rear sprocket of 3.6 inches. If the rider can maintain a maximum pedal pump rate of 42 pumps per minute on level ground, and the wheels on the bike have a radius of 27 inches, what is the maximum speed the rider can attain?

A hot air balloon leaves the ground and climbs vertically. After 3 minutes pass, an observer half a mile from a point directly below the balloon must lift her gaze from vertical through an angle of 12 degrees to look directly at the balloon. How high is the balloon?

The length of a property line needs to be precisely measured, but there is a huge rock, roughly 9 feet in radius and 15 feet tall that sits squarely on the line. Short of raising a measurement tool above the height of the rock, how can an accurate measurement be taken?

A contractor has been hired to construct a wheelchair ramp. The ramp's inclination may not exceed 3 degrees. The ramp must rise from a point on the ground to a porch that is 5 feet off of the ground. If the cost of wood will be approximately 12 dollars per foot of ramp, how much will the wood cost?

Finite Mathematics and Applied/Business Calculus

You are concerned about having enough to money to send your child to college. If he is currently 8 years old, and you believe that the market will consistently return at least 9% on investments, how much money should you invest monthly to guarantee that you will $40,000 in your child's college in 10 years?

One bank offers 3.25% interest on regular checking account, compounded quarterly. Another bank offers 3.2%, compounded daily. If you are depositing a fixed amount of money for one year, which is the better deal?

You are interested in buying a house that sells for $90,000. You are able to secure a loan for 95% of the amount required at 8.7% interest, financed for 30 years. After paying the $4,500 down payment, you determine that you can afford a monthly payment, before taxes and additional charges of $825 per month. Can you afford this house?

Statistics

In a random sample of 620 likely voters, 335 people favor Annie for public office. Is this enough support for the candidate to assume that she will receive a majority of voters in the full election?

You wish to determine the time required to get from your house to a local hospital. In an informal survey of 11 of your neighbors, they report that they can get to the hospital in 8, 11, 15, 10, 7, 7, 12, 10, 8, 6, and 13 minutes respectively. Does this sample assure that you can expect to get to the hospital in less than 15 minutes?

Probability

Customers arrive randomly and independently at a post office at a constant rate of 2 per minute. Three postal agents are available to serve customers at the counter and the average service time per customer is 1 minute. Customers arriving at the post office form a queue and wait in turn for the first available postal agent. There are currently 5 people in the queue waiting to be served. What is the probability that a person just entering the post office will have to wait more than 5 minutes to be served? What is the expected length of the queue 5 minutes from now?

The risk of failure for a particular brand of tire with x miles of travel wear is proportional to x^2 (i.e., x-squared). There is a 3% chance that a new tire of this brand will wear out in the first 15,000 miles of use. What is the probability that a tire with 5,000 miles of wear will last for another 15,000 miles?

The value of a particular investment fluctuates from day to day, sometimes increasing and sometimes decreasing. On 60% of the days, the investment increases by 1%; on the other 40% of the days, the investment decreases by 0.5%. Daily returns on the investment are independent, which means that the pattern of daily returns prior to any given day has no impact on the return for that day. What is the probability that this investment will be worth double its current amount one year from today? How long should we expect to wait until this investment doubles?

Under the terms of a particular comprehensive medical insurance plan, an insured person receives a 80% reimbursement of eligible medical expenses after paying a deductible of $500 and receives 100% reimbursement once total out-of-pocket expenses reach $1500. In any given year, the total eligible medical expense is $200, $800, $1500 or $10,000 with frequencies 40%, 30%, 20%, 10% respectively. An insurer provides coverage of this type to 1000 people. What is the probability that the insurer's total payments under this plan exceed $1 million? What is the expected reimbursement per person?

The drivers insured for collision damage by a particular auto insurance company fall into one of two classes: average or high-risk. Seventy percent of drivers are considered average. In any given year there is a 1% chance that an average driver will submit a claim and a 5% chance that a high-risk driver will submit a claim. When a claim is submitted, it is of size $1000, $5000 or $20,000. Fifty percent of the claims submitted by average drivers are of size $1000, 40% are of size $5000 and 10% are of size $20,000. For high-risk drivers the corresponding percentages are 20%, 45%, 35%. What is the expected payment to a driver whose risk class is not specified? What is the probability that the actual payment exceeds this expected amount by more than $1000?

Mathematics for Biological Sciences

Calculus and differential equations can be used in population models to determine population size, or population rate of growth or decay.

Models based on simple trig functions can be used to build simple models for studying the workings of various biological systems. As a simple example, a damped sine function can be used to model the rate of heartbeat of an animal that has gone from strenuous activity to a resting state.

Calculus

A statistical procedure called regression can be used to determine the equation for a function that closely approximates the curve of an object or machine part. Calculus techniques can then be used to determine the linear length along the curve, the center of mass of the object bounded by the curve, and many other useful quantities.

Some simple applications of calculus include the area of a flat surface, the surface area of a three dimensional object, the volume of a three dimensional object, the mass of a three dimensional object (given its density), water pressure at different depths, air pressure at different heights, and probabilities of complex events.

Differential Equations

Differential equations are frequently used for simple population growth rate questions. It is now fairly common in elementary differential equations courses to examine what is known as a predator-prey model of the interaction of species.

Differential equations can be used to model the motion of parts of machines, including pistons, gears, pulleys, and springs.

Differential equations may be applied to three-dimensional surfaces to determine paths of steepest ascent and descent.

Current across an electrical circuit can be modeled with differential equations.

Differential equations can be applied to the periodic measurement of tumor size to determine the rate of growth, and thus the severity of cancerous growth.

Linear Algebra

Linear algebra is used to resolve problems associated with large-scale systems of linear equations. Such equations may arise from a number of applied fields, including business and sciences.

One aspect of linear algebra deals with manipulation of large sparse matrices. Sparse matrices are matrices that contain a relatively large number of zero entries. Such matrices are important to the analysis of problems in graph theory, including large-scale task scheduling.