Among all pairs of numbers whose difference is 14, find a pair wh...

Question

Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

Accepted Answer

Hello. Today we're going to find a pair of numbers whose difference is equal to 22 and whose product yields a minimum value. So let's just go ahead and take this problem piece by piece first. We are told to find a pair of numbers. So that means that we're gonna be working with a number X and a number Y. And the reason why we're labeling them as variables is because we don't know what those numbers are Next. We are told that the difference is 22. So we know that the difference of these two numbers is going to produce the value 22. Well, let's just go ahead and assume that Y is going to be the bigger number in this case. So Y -X is going to equal to 22. And we are also told that the product of these two numbers is going to be a minimum value. So that means that the product X times y is going to give us an unknown value. So our job is to figure out what that value might be. Well, how do we go about starting this? Well, we have two equations to work with why minus X is equal to we have X times Y. Well, we can go ahead and create a substitution for why using our first equation. So in order to make a substitution for y, let's go ahead and add X to both sides of this equation. Doing so is gonna give me Y is equal to X plus 22. Now what I can do is go ahead and take this substitution and plug it into the product of this of X times Y. Doing so is going to give me X times the value X plus 22. And in order to simplify this, we're gonna go ahead and distribute X into this parentheses and that's gonna give us x squared plus 22 X. Well, what does this do for us exactly? What does this tell us? Well, if you notice this roughly resembles the equation of a positive parabola and it's a parabola because the leading power is to so this is a function of a parabola. The parabola is positive because the leading coefficient is positive and that's actually a really good thing because if we have a positive parabola, the very smallest value or the very minimum value of that parabola occurs at the vertex. So if we could find X and Y of the vertex then we can take the product of those two numbers and get the minimum product. So how do we start that? Well, let's go ahead and recall how we find the X value of the vertex, the X value of the vertex of a. Parabola is negative. B over two. A. And we need to go ahead and identify what are B and A. Is in this case while B is just the coefficient in front of our X term, which in this case is going to be 22 and A is going to be the coefficient in front of our X square term. And normally there's nothing written here, but this is really one times X squared. So that means that A is going to equal to one. So if we take these two values and plug it into the equation to find X of a vertex, we can get the minimum X value which is one of the numbers we're looking for. So X is going to equal to negative 22/2 times one, Two times one is just gonna be too. So we're left with negative 22/2 and negative 22/2 is going to produce the value of negative 11. So what this means is that X is going to equal to negative 11. So now we have a minimum X value. Well, what can we do with this to try to get a minimum y value? Well, we do have an equation that can help us solve for a minimum y value. And it's the equation that we saw for earlier, the substitution for y, Y is equal to X plus 22. So if we take our value of X and directly plug into the substitution for Y, we're gonna go ahead and get a minimum value for y. So let's go ahead and do that. Why is going to equal to negative 11? Because we're plugging in the value for X that we found plus 22. That means that negative 11 plus 22 is going to give us positive 11. So why is going equal to positive 11? So now we found our values for X and Y. What do we do with these two values? Well, remember we're asked to find what the minimum product is and we know that both are X and Y, values are minimum values. So if we take the product X times Y and we plug in X times Y, which in this case is going to be negative 11 times 11. This product is going to produce a minimum value or the minimum product we're looking for. So negative 11 times positive 11 is going to give us -121. And that's gonna be our minimum product. That also means that the solution to this problem is going to be B. So, I hope this video helps you in approaching this type of word problem. And I'll go ahead and see you all in the next video

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