Using Math in the Real Life

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Math is a powerful tool for resolving the questions that life sends our way. We can find the best price, calculate a particular distance, or even use probabilities to make decisions. In this lesson we will explore some ways to use math in real life.

Mathematics as a Tool

You're enjoying the view from a small plane window, 10,000 feet above the earth, when your friend Susan suddenly shouts ''Bring me a parachute!'' and steps out of the plane. How much time do you have to save Susan?

Mathematics is the study of numbers as they relate to equations, functions, and shapes. It's the language that describes the framework of reality and the patterns of existence. Learning mathematics allows us to work with and investigate reality.

So, let's save Susan and go through a few examples of using different kinds of mathematics to solve real-life situations.

Saving Susan

Using algebra, we discover how long it will take poor Susan to reach the ground. Susan's velocity will increase according to the pull of gravity (32 feet per second faster every second) until she reaches terminal velocity, which is about 176 feet per second if she's in the skydiver spread-eagle position to catch as much air resistance as possible.

Airplane Problem
free fall correct

First, we have to find how long it takes Susan to reach 176 feet per second and what her height will be then. To find that time, we can divide the terminal velocity by her acceleration (how fast her velocity is changing):

terminal velocity time

Okay, but how high is she when she reaches terminal velocity? Well, her height is changing by -(1/2)(32 feet/second²)t². We use -1/2 * acceleration * time² to average the beginning and end velocities, since Susan's velocity is changing over that time. Since she's falling from 10,000 feet, we get this expression:

first falling time

If we put in 5.5 seconds for t1, we find Susan at h1 = 9516 feet when she reaches terminal velocity. At terminal velocity, air resistance will slow her down as quickly as gravity is trying to speed her up, so she'll be falling at a constant velocity for the rest of the time. This makes finding t2 straightforward, since we already know that she's now falling from a height of 9516 feet:

second falling time

You can divide Susan's height by her terminal velocity to find that t2 = 54.07 seconds. That plus the 5.5 seconds it took her to reach terminal velocity gives her 59.57 seconds before she reaches the ground.

Poker Odds

Royal flush is the highest hand in poker. What is the probability?
royal flush image

Poker is a card game where particular combinations of cards have higher values than others, in a deck that has 13 different card values and four different suits, for a total of 52 cards. Probabilities can tell us how likely it is that you'll get a winning hand in card games. So, what is the probability of getting a pair, two cards with the same value? If you divide the possible successes in the deck (the number of cards that are ones you want) by all of the possibilities that could happen (all of the cards in the deck), you'll have a single-event probability. When you're dealt five cards you have five events that will happen in a row and will each have a different probability.

Your first card will be one of 13 values and one of four suits. You will want to match the value, so if you get a 6 for your first card you'd like to get another 6 on the second card. The second card is dealt from the 51 remaining possible cards. Three cards in the remaining deck are 6's and will make a pair. If you divide 3 by 51 (3 successful results divided by the 51 cards in the deck), you get a little less than a 6% chance. Not great, but you still have three more cards coming.

If you didn't get a pair in the first two cards, there is a 6/50 chance of getting a pair on the 3rd card, because there are three cards that match your first card and three that match your second one. In the same way, your 4th card has a 9/49 chance, and your 5th card has a 12/48 probability that you will get a pair.

This is called an OR situation because you're hoping that your second card OR your third card OR your 4th card OR your 5th card will make a pair. So far, you're only calculating the probability of each event, but is there a way to find the total probability?

The easiest way to determine the overall chance of getting at least a pair among the five cards in your hand is to calculate the chance that you won't get a pair and then subtracting that result from 1, since the probability of something happening plus the probability of something not happening equals 1 (100%).

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