There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue. First, we rewrite one variable in terms of the other:. The graph of this quadratic function opens upwards, and its vertex is the minimum, So if we find the vertex of this parabola, we will find the minimum product. The vertex is:. A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard.
She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side. It is also helpful to introduce a temporary variable, W , to represent the width of the garden and the length of the fence section parallel to the backyard fence. Now we are ready to write an equation for the area the fence encloses.
We know the area of a rectangle is length multiplied by width, so. The function, written in general form, is. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers.
This is why we rewrote the function in general form above. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.
This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function below. The unit price of an item affects its supply and demand.
That is, if the unit price goes up, the demand for the item will usually decrease. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue? Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity.
Because the number of subscribers changes with the price, we need to find a relationship between the variables. From this we can find a linear equation relating the two quantities. If you choose 60 for the internal angle of an equilateral triangle you get degrees in a circle. The radius of a circle will fit inside the circle six times exactly to form a hexagon; the corners of the hexagon each touch the circumference of the circle.
Babylonians did indeed have a love for the number 60 and if each of the sides of the hexagon are divided into 60 and a line drawn from each 60th to the centre of the circle then there are divisions in the circle. Thanks for going to the trouble of explaining the history and applications of quadratic equations. The point of it all was never explained to me when I was thrown into the deep end with them, age Now that I've been asked to explain them to a friend's son, your material is helping to demystify things.
Matt, North Wales, UK. How is this equation derived from the figure given? There's no explanation as to what "a" and "b" actually represent? I was wondering the same thing. In the diagram I take ax to be the base of the smaller triangle but then where is x in the equation coming from? Are a and x equal? I'm also stuck on that 1st example of the field comprising 2 triangles and how we get to the quadratic equation from that. I would love to go through the rest of this article but don't want to until I've overcome the hurdle of understanding this.
Please, someone? But why is the base of one triangle ax and of the other simply b. Where does that ax value come from? I can understand Anon's frustration back in Jan ' So often in mathematical explanations I've read I find myself tripping over a missing step.
Like a mathematical pothole. It's usually something so obvious to the mathematician who wrote it that it didn't seem to need mentioning.
Like where that little square came from- though I did eventually work that one out. The problem is that if you are trying to follow a set of mathematical steps even if you solve the missing one as with me and the small square you have been diverted away from the main problem and lost the thread: And then probably give up and go off and do something else instead.
I'm pretty sure when they sought out ways to derive a quadratic equation to help them reason triangular regions they had to think frontwards and backwards. First, keep in mind that "m" represents a basic unit of 1. But are their heights 2x? Let's think about it, when finding the area of a right triangle we eventually divide the area by 2 after multiplying the b x the h. The larger triangular area would be b times x or bx for its area.
You asked though "what is "a" and "b"? This is my perception after being confused there for a minute too. I hope this helped you or someone just a little although it's years later- just discovered this awesome forum:. I really can't follow what you're saying. I just want to know where that expression for the height comes from.
So the yield, which should be a product of area and the coefficient m is now rendered as the areas of two squares without having anything to do with that coefficient anymore. I can see all that but I just can't grasp what on earth is going on and its doing my head in. Babylonians took over Mesopotamia at around BCE.
Thanks so much I kept getting my anwsers wrong because I didn't realize you had to divide both parts by the denominato. Allaire and Robert E. I noticed a few people were confused about the choice of height for the triangle, so here is my explanation : m is the amount of crops that you can grow in 1 square unit of area.
Skip to main content. Chris Budd and Chris Sangwin. March Babylonian cuneiform tablets recording the 9 times tables. Sunflower seeds, arranged using Fibonacci numbers. The Parthenon, embodying the Golden Ratio. A cross-section of a cone can be a circle And also the description Permalink Submitted by Anonymous on January 17, Permalink Submitted by Anonymous on November 25, Yes you are right, they weren Permalink Submitted by Anonymous on February 23, Yes you are right, they weren't around yet.
Permalink Submitted by Anonymous on January 27, Very Cool Story! Permalink Submitted by Anonymous on May 3, Triangular field area Permalink Submitted by Anonymous on October 10, Great article, wonderful introduction to quadratic equations. Re: Very Cool Story! Thanks for your rectifying. Permalink Submitted by Gregory D. Appreciation Permalink Submitted by Anonymous on September 16, Wonderful article!!!!!!!!!
More generalized polynomials can be a pain to factor, though. Well, okay.. Permalink Submitted by Anonymous on January 13, Bad teachers Permalink Submitted by Anonymous on June 28, So, to decide where to set your price, use P as a variable. You've estimated the demand for glasses of lemonade to be at 12 - P. Using however much your lemonade costs to produce, you can set this equation equal to that amount and choose a price from there.
In athletic events that involve throwing objects like the shot put, balls or javelin, quadratic equations become highly useful. For example, you throw a ball into the air and have your friend catch it, but you want to give her the precise time it will take the ball to arrive. Use the velocity equation, which calculates the height of the ball based on a parabolic or quadratic equation.
Begin by throwing the ball at 3 meters, where your hands are. Also assume that you can throw the ball upward at 14 meters per second, and that the earth's gravity is reducing the ball's speed at a rate of 5 meters per second squared.
If your friend's hands are also at 3 meters in height, how many seconds will it take the ball to reach her? The method is explained in Graphing Quadratic Equations , and has two steps:. Yes, a Quadratic Equation.
Let us solve this one by Completing the Square. Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:.
The frame will be cut out of a piece of steel, and to keep the weight down, the final area should be 28 cm 2.
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