Problem

A window panel for a sun porch consists of a 7 -f…

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Problem 39 Medium Difficulty

An architect has allocated a rectangular space of $264 \mathrm{ft}^{2}$ for a square dining room and a $10-$ ft wide kitchen, as shown in the figure. Find the dimensions of each room.
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Answer

Dining room: $12 \mathrm{ft}$ by $12 \mathrm{ft} ;$ kitchen: $12 \mathrm{ft}$ by $10 \mathrm{ft}$

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Algebra

Elementary and Intermediate Algebra

Chapter 6

Polynomial Factorizations and Equations

Section 7

Applications of Polynomial Equations

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Video Transcript

in this particular problem, we are, um, given a rectangular space, the site of size of 264 square feet, that rectangular space is going to be a kitchen and a dining room. But the question is what will be the dimensions of either one? So we have to read further into the problem. And when you read further into the problem, they have divulged that the dining room is square. It says so. So I am going to use that rectangle and know that this will be my dining room. He looks pretty darn square to me. So there is my dining room, and what's left will then be the kitchen. So I'm gonna color code these a little bit just to clear it up a little bit. Let's call my read the dining room and my green. The kitchen. The kitchen is not necessarily square. We don't really no if he's rectangular or square. But we do know the dining room. Miss Squid. What we do know about the kitchen is that it is 10 feet wide, so this green right here is going to be 10 feet wide. So on either side of the rectangle I'll give it a mark of 10 feet, but what are the dimensions of everything else? So we have to bring in that variable the thing we're trying to find. We are going to substitute or not. Substitute called the size of the dining room acts. Now, since he's a square, we know every single side of him. Once I call one side X, all other sides become acts. So now, if you look at the diagram, you realize that this side of the kitchen will also be X, and this sign will be X. So we have all four sides of every triangle, every I'm sorry rectangle or square, given a certain variable or number. The 10 was given the X we created to find the rest of the dimensions. So now let's look at this closely and figure out the area of the dining room based on acts and the area of the kitchen. We know the area of a rectangle or a square is base times, height length, times width. The dining room is length times with and it ISS X squared. That is the area, uh, my kitchen. So the area is X squared to do area to do areas X squared. Now what's the area of the kitchen again? Length. Times, width of our base, times height for rectangle. 10 times acts. So he is simply 10 X. So area is 10 acts. So now we know the areas. OK, so now let's look back at the problem and see what other information they've given us. They have sad the combination of the two areas. The dining room plus the kitchen will total 264 square feet. So there we go. The label of feet squared dropped the label For now. Don't let this square in the label confusion. We are not squaring any numbers. Feet squared is a label. It's a name. It's not part of a calculation. So now let's substitute dining room. We found to be ax squared. That was its area. We noticed the area of the kitchen is 10 X, and the area of them combined is 264. So now I'm looking at this and getting ready to solve for X. So one thing you notice is my highest power of X is a square that makes this a quadratic that makes this graph into a parabola, and it makes the means of solving become factory. Factoring means we must make one side equal to zero, and all other terms be put on the other side of the equal sign. So in this case, the easiest one to move is the 264. So I'm gonna take that 264 and send him to the other side through subtraction, because 264 minus 264 0 And that's what I want over there on the right, and I'm gonna also subtract them from this other site. I'm going to sneak him in right here after the 10 ax because he's a constant. He belongs at the end, and he can't combine with any life terms on the left. So what I have here is X squared can x in the minus 2 64 and that illegal zero. So because I have a positive one x squared, my life has made a lot easier The minute I put a negative in front of that X squared or a coefficient, that's not positive. One. This is a bigger headache. This is not as big of a headache to work with because it's positive X Square. So as soon as you know I'm sorry. Positive one X word As soon as that's the case, this is what you will know. This will break itself down. We hope into two factors with X leading the way in both equal to zero. Now, if it's era an irrational solution, it's not this easy. But when it's a rational solution, we definitely know when it's a positive one in front of the X squared that this is how it's gonna look. To start with the rest. We have to know, figure out. We have to ask ourselves what multiplies to 264. That also adds to 10. And when you think this through when you're had, want multiplies to negative 264. That also adds to town. It is negative. 12 and positive. 22 positive 22 times Negative. 12 This negative 264 negative Positive. 22 plus negative. 12 This positive 10. So we have made this work for us. Now what we all now We're going to apply the zero product property which states if you have factors equal to zero all it takes is one of those factors equaling zero to make this a true statement. So here is my first factor, X minus 12. If I could make X minus 12 equal zero, it doesn't matter what any of the other factors are on the left side, this whole left side will become a zero. So I'm gonna, um, work through that scenario if my first factor is your Oh, that's what it I just stated right here. Then so is the entire left side. So when I add 12 to both sides of my equal sign, I will now see that my first solution is X equals 12. I say my first solution because there are other factors on the left side that are gonna produce solutions as well. One of the other factors and the only other factor here is my second factor is X plus 22 zero product property says now it's that one's a zero. It doesn't matter what X minus 12 is. This whole left side will become zero and make a troop. So my other scenario, when I subtract the 22 says my other solution is potentially X equals 22. These aerial solutions. They are true solutions. But what makes minus 22 a bad solution? I'm measuring a dining room in a kitchen. What is negative? 22 feet. It means nothing. We cross it off and say it. It does solve the problem, but it doesn't work in this word problem. What works is X equal 12. So that's the solution that we're gonna work with now. We're gonna jump back up to the problem. And wherever we see an axe, we're gonna insert 12 feet. So we now see that the dimensions of the dining room are 12 feet by 12 feet NYT 12 feet by 12 feet. It's 12 all around because it's a square. All of these X is now taking on a 12 and same with the green. There are exits. They deserve the right to be called 12 as well. So if you are answering this word problem in a complete sentence where the question does ask, find the dimensions of the rooms. You would say the dimensions of the dining room is 12 feet by 12 feet, and the dimensions of the kitchen are 12 feet by 10 feet. Beautiful

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