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Applying Quadratic Functions to Motion Under Gravity & Simple Optimization Problems

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  • 0:03 Quadratic Functions
  • 1:32 Real World Problems
  • 6:33 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Quadratic functions are very useful in the real world. Watch this video lesson and learn how these functions are used to model real world events such as a falling ball.

Quadratic Functions

In this video lesson, we will talk about how quadratic functions, the function of a degree of 2, are used in the real world to model real-world scenarios. Remember that a function with a degree of 2 has, at most, three terms with the highest exponent of 2. All of our quadratic functions follow the standard form of f(x) = ax^2 + bx + c, where a, b, and c are your coefficients and x is your variable.

When used to model real-world scenarios, though, your variable can be another letter; it is not always x. It can be t to stand for time, or h to stand for height, or any other letter to stand for whatever is needed by the problem. For example, this quadratic function is the basic function for an object that is either thrown up into the air or just let go:


quadratic real world functions


The g comes from the force of gravity. It is either 4.9 in meters or 16 in feet. The v sub 0 stands for the initial velocity of the object, and h sub 0 is the height from which the object is thrown up or dropped. Notice the exponent of 2.

Our formulas for calculating area are also quadratic functions. For example, the function to calculate the area of a square is A = s^2, where A stands for the area and s stands for the measurement of a side of the square. Do you see the exponent of 2?

Real World Problems

What kinds of real-world problems can we expect to solve using these quadratic functions? We can solve problems where we want to find how much time a certain object stays in the air before hitting the ground after being thrown into the air, or dropped, or thrown down from a certain height.

For instance, say someone was throwing candies from the third floor. You are on the bottom and want to know how long you need to wait before putting your bucket out to catch the candies. You can use a quadratic function to find out.

And we can also solve problems like finding the best dimensions for a yard to hold the largest area given only a certain amount of fencing. Say you were given only one roll of fencing. You can use a quadratic function to help you find the largest area you can enclose with that amount of fencing. You want to see how this works with a couple of examples?

Okay! Let's talk about the candy one first. This deals with gravity, so our function is based on how objects fall when under the force of gravity. Living on this planet Earth, you know that everything is affected by the force of gravity, which happens to be the same for all objects. Therefore, we have this formula for objects falling under the force of gravity:


quadratic real world functions


The little g, which is derived from the mathematical value of gravity, is 4.9 if we are dealing with meters and 16 if we are dealing with feet. Here is our problem:

'The candy man is throwing candies out the third floor window of his candy factory. He throws the candies up into the air at a velocity of 16 feet per second. The height of the third floor window is 32 feet. How many seconds does it take for the candies to hit the ground?'

We see that this problem gives us our v sub 0, our initial velocity, as well as h sub 0, our initial height. Since we are dealing with feet, our g is 16. Plugging in our values of g = 16, v sub 0 = 16, and h sub 0 = 32 into our gravity function, we get this formula:


quadratic real world functions


This function gives us the height of our object, our candies, at any given point in time. To find when our candies will hit the ground, we set this function equal to 0 for a height of 0:


quadratic real world functions


We then use algebra skills to solve this quadratic function. We first divide by -16. We get 0 = t^2 - t - 2. We can then solve by factoring. It becomes 0 = (t - 2)(t + 1).

We get answers of t = 2 and t = -1. Which one of these answers makes sense for us? Well, we can't go backwards in time, so t = 2 is the answer that makes sense. It takes 2 seconds for the candies to hit the ground.

Our next example takes a look at optimization, or making the most of a situation. Our problem is this:

'We have 500 feet of fencing available for us to use. What are the dimensions of the largest size rectangular yard this amount of fencing can make?'

To solve this problem, we need to make use of the area formula for a rectangle. The formula is A = L * w. We have two variables. To turn this into a quadratic function, we need another equation that we can solve for one of the variables to plug it into the area formula.

This second equation we will get from our 500 feet of fencing. Our formula for our perimeter is 2L + 2w = 500. Solving this w, we get w = 250 - L. Plugging this into the formula for area, we get A = L * (250 - L) = 250L - L^2 = -L^2 + 250L.

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