# When Will I Ever Use This Math?: Real-World Math Applications

By Eric Bias, Layout and Sports Editor

The question that acts as this article’s headline is one that I hear all of the time by my peers in math class. Even though this is a simple and valid question to ask, I still find myself getting annoyed when one of my peers asks it. Despite the fact that I don’t know where the specific kind of math that we are learning at a certain time is used in the “real world”, I know that it is used. With the world economy demanding more and more science and math majors, it is important that students understand that math is used universally and it is the language of science. Science and mathematical knowledge is the path to a better future.

Listed below is an outline of all of the topics that are covered in high school, from grades 9-12, and where they are used in the “real world”:

9th Grade (Algebra): Algebra is the math that is, arguably, the most applicable to your everyday life. We use algebra while we go to the grocery, go to the bank, and even drive on the road.

Expressions: Algebraic expressions are expressions that are constructed from integer constants, variables (like X and Y) and algebraic operations, such as addition, subtraction, multiplication, and division. Expressions are used, for example, when you are at a car rental agency and they have a flat rate, along with an hourly rate to rent a car. An example of an expression that could represent the cost to rent a car could be represented by: y = 45x + 200.

Equations and Inequalities: Equations are statements that evaluate two expressions as they are equal in value. Inequalities show the relationship between two values that are not inherently equal. Inequalities are used in the real world to relate the values of two unequal values. For example, we can relate the value of a car to the value of a bicycle.

Polynomials and Rational Functions: Polynomials are expressions that consist of more than two algebraic terms. Rational functions are any functions that can be defined by a rational fraction (this is where both the numerator and the denominator are polynomials). Polynomials are used to model stocks on the stock market to show how their value varies overtime.

Congruency: The word congruency means “exactly equal in size and shape”. When shipping companies like UPS try to fit as many packages as they can on each semi truck, they utilize the idea of shape congruency in order to maximize the number of packages they fit per cargo shipment.

Right Triangles and Trigonometry: Right triangles are triangles that have one 90 degree angle. Trigonometry is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles. Trigonometry is used heavily in the field of engineering. Right triangles are used in bridges, since they are the strongest shape that can support a bridge or related structure.

Circles: Circles play an essential role in science and math careers. Circles are the representation of all points that are the same distance (equidistant from the center). Knowledge of circles helps in many different fields of study. For a real life example, let’s suppose that you are watering your lawn with a singular sprinkler. In this scenario, it would be important to adjust the sprinkler to encompass as much of the lawn as possible. Understanding circles and how they are constructed can help you solve your watering problem.

Modeling:  A branch of applied mathematics and computational geometry that studies methods and algorithms for the mathematical description of shapes. Astronomers use geometric modeling to try to represent the shapes and forms that stars, planets and galaxies take. Their modeling helps us gain a better understanding of the universe in its entirety.

Rational Functions: Rational functions are those functions that have both the numerators and denominators are polynomials. For example, let’s say there are two numbers. The second number is 1 larger than twice the first number. The sum of the reciprocals of the two numbers is 7/10. Find the two numbers. Rational functions can help you solve this problem.

Trigonometric Functions/Equations: A function of an angle, or of an abstract quantity, used in trigonometry, including the sine, cosine, tangent, cotangent, secant, and cosecant, and their hyperbolic counterparts. If you listen to a weather forecast to see what the weather’s going to be, that depends on trigonometry. Meteorologist use trigonometry by using data that they gain from weather balloons suspended thousands of feet in the air in order to accurately predict weather patterns.

Function: Functions show a relationship involving two or more variables. The temperature over time for a certain city is an example of how functions are used in real life. This function could help residents predict the temperature of their city at some point in the future.

Sequences and Series: Sequences are enumerated collections of objects in which repetitions are allowed. Series is the sum of the terms of an infinite sequence. An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats. How many seats are in the theatre?

Vectors: Vectors have magnitude and direction. In math, a vector can be thought as a line segment where the length of the arrow indicates its magnitude and its direction is measured from its head to its tail. An example of how vectors are used in real life is when airplane pilots receive landing directions from Air Traffic Control. An even more relevant example of vectors being used in real life if when you drive in your car for a set distance and a direction (i.e. 4 miles northeast).

Rates of Change: Rate of change in math describe the derivative (the slope of the line represented on the graph) with respect to time. For example, when Brian turned 18, his parents helped him buy a car. If he starts off owing his parents \$500, and can pay back \$150 every two weeks by working at his job. How many weeks does it take for him to pay off his parents? What is his rate of payoff per week?

Limits at a Point/Limits involving infinity: Limits describe the value that a function or sequence approaches a specific x-value. Let’s say that you have a chemical reaction in a beaker start with two chemicals that form a new compound over time. This could be represented by a limit problem.

Derivative/Derivative Rules: A derivative is a way to represent the rate of change on a graph. If you want to know how something changes over time, you usually need to use a derivative to understand how it changes.

Trigonometric Functions: Trigonometric functions are used very heavily in calculus. Architects and engineers use trigonometry and trigonometric functions to understand how to construct their desired structures.

Extreme Values: When an output value of a function is a maximum or minimum over the entire domain of the function. Extreme values can be used by businesses to determine what the minimum price that a good or service can be sold at and still make a profit.

Optimization Problems: Optimization is the greatest or least value of a function for some constraint. For example, A rope is strung from the tops of two vertical poles. Between the poles, it is tied to a point on the ground. Show that the shortest length of rope occurs when the two angles that the rope makes with the ground are equal.

Linearization Models: Finding the linear approximation to a function at a given point. Initially, trains A and B are  miles away from each other. Train A is traveling towards B at  miles per hour and train B is traveling towards A at  miles per hour.  At what time will the two trains meet? At this time how far did the trains travel?

Related Rates: Related rates involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. A ladder 20 feet long leans against a building. If the bottom of the ladder slides away from the building horizontally at a rate of 4 ft/sec, how fast is the ladder sliding down the house when the top of the ladder is 8 feet from the ground.

Euler’s Method: A first-order numerical procedure for solving ordinary differential equations with a given initial value. Euler’s method could be used to estimate the national debt in ten years.

L’Hospital’s Rule: This is a rule that is used to evaluate indeterminate forms of limits. L’Hospital’s Rule is used to prove that the compound interest rate equation through continuous compounding equals Pe^rt.

Definite Integrals and Antiderivatives: Antiderivatives are the inverse operation of differentiation. For example, definite integrals are used at the beach. Let’s suppose that a bucket has an empty weight of 23N. It is filled with sand of weight 80N and attached to a rope of weight 5.1N/m. Then it is lifted from the floor at a constant rate to a height 32 meters above the floor. While in flight, the bucket leaks sand grains at a constant rate, and by the time it reaches the top no sand is left in the bucket. You can use integrals to find the work done by you on the bucket.

Integration: A mathematical object that can be interpreted as an area or a generalization of area. The vertical end wall of a trough is in the form of an isosceles triangle with one the point down. The top of the wall measures 4 units wide and the depth is 6 units. Find the pressure force on the wall caused by the liquid in the trough when the trough is just full to the brim.