the-perimeter-of-a-rectangle-is-24-feet-a-if-x-is-the-measure-of-one-side-of-the-rectangle-represent

Question

The perimeter of a rectangle is 24 feet.
a. If $x$ is the measure of one side of the rectangle, represent the measure of an adjacent side in terms of $x .$
b. If $y$ is the area of the rectangle, express the area in terms of $x$ .
c. Draw the graph of the function written in b.
d. What are the dimensions of the rectangle with the largest area?

Finian Lickona · Accepted Answer

Okay, In this exercise, we are given that the perimeter of a rectangle is 24 feet. So that means if we add up the length of this side, this side, this side, this side get that the perimeter equals 24. So the first part of the question says, if X is the measure of one side of the rectangle expressed the length of the other side in terms of X. So let&#x27;s think about how this breaks down. So if this is X, then this means this side length is also X. And we know that if we add them together and then add in the length of the other two sides, we&#x27;re just gonna call those l we get plus two l equals 24. So we can actually, uh, find this l value in terms of X. So the first thing I&#x27;m gonna do is subtract two X from both sides of the equation. We have to l equals 24 minus two x. And if we divide away the two from every single term in the equation, we get l equals 12 minus X. Okay. So we can actually I&#x27;m gonna go ahead and erase all of that. That&#x27;s not gonna work. Okay, We can actually rewrite l the quantity l as just being 12 minus x. Okay, so let&#x27;s the first part of the question. The second part is, if why is the area of the rectangle express the area in terms of X? All right, So area, the area for a rectangle is given by base times height. So in this case, they&#x27;re height is X, and our base, we just calculated is 12 minus X. So if why&#x27;s the area well yet why equals x times the quantity 12 minus X. And if we expand that out in rearrange terms, we get negative X squared plus 12 x. Okay. And now we want a graph of this function, which I had already generated. But this is what the parable, a negative X squared plus 12 x looks like if you graph it on a set of axes and we need Thio for the last part of the question, find the dimensions of the largest rectangle. So this parable represents the area for each measurement value of X. So it looks like the area is at its greatest at 36 and that makes sense, because if we have a perimeter of 24 the highest area quadrilateral we can create with that is just a perfect square. And if we do that, we get that our area six times six is 36. If we were to do different permanent measures like, uh, 10 and wrong spot 10 and two, we would get that the area is 10 times two is 20. So, uh, 36 is clearly about as big as it gets and

Joshua Utley · Answer

all right, we&#x27;re asked first to represent the perimeter of this rectangle that I&#x27;ve drawn, using the information given using in terms of X. And so the perimeter is just all the sides added up, so I&#x27;ll have X plus two X plus five plus X again and and plus two X plus five again. So you have the length twice in there, and you&#x27;re the width twice. What that&#x27;s going to give you is you got to X there to have one x three ex Florence six X and then here I have the time. All right, so that&#x27;s part a six x plus 10 and then Part B says if the perimeter is equal, thio two feet more than eight times the width, which should record, which you would write is two feet. Waas eight x two feet more than eight times the width. Find the dimensions of our rectangle. So the idea is here is now we have two equations that represent our perimeter. Here we have one that says six X plus 10 and one that says two plus eight X. And since they&#x27;re both representing the same thing, these are also both equal and so What I can do here is I can take six x from this side in six sex from this side and I&#x27;ll be left with two plus two. X is equal to 10 subtracted, too. Load sites and I will have that to X is equal to and which one&#x27;s X is equal to four crapped. If I just solve for the X here. Well X, as we put said before appear, represents a bullwhip, so that&#x27;s part of it. And then, if we bug into two X plus five here, two times four is eight plus five, which is 13. So this rectangle is a 13 loan 54 wide rectangle.

Emma Chickles · Answer

this question first asked us to describe the area of a rectangle. Why, as a function of its length X, we know if the perimeter of the rectangle is equal to 12 so we can describe the area wide by introducing a new variable L as the other is, set the length of the other side. The rectangle, where two times X plus two times l is equal to 12. Rearranging this equation for l gives us that two times l is equal to 12 minus two times x or dividing this equation by two l is equal to 61 sex. Now we confined area this rectangle. Why? Why is equal to the, um, the product of these two sides? X times l Substituting six minus x in for l gives us that y is equal to x times six minus X Paying us and set builder notation. This gives us that why is equal to X times six mine sex Now I asked if solve this equation for different values of X from 0 to 10 when X is equal to zero, why is also equal to zero zero time zero when X is equal to one? Why is equal to one times five and so on. To save time, I&#x27;ve written out the next nine values from 2 to 10. Now we can use these values to plot the graph of this Kurt. Also note that for X more than equal more than six. This gives negative values with the area of which doesn&#x27;t make physical sense, but we&#x27;ll accommodate for that in the domain in part. See, for now, let&#x27;s block this graph. This graph goes to the origin. It has another exit except at X equal six and has the Maxwell Med X equals three and Michael tonight. So the scarf looks like this. Lastly, we&#x27;re asked to find the domain of this graph, the smallest value that makes physical senses zero and largest value is six. That&#x27;s because beyond six, the area&#x27;s negative. And so this doesn&#x27;t make any physical sense. So the domain is older. Male numbers from 0 to 6

Noraney Ocampo · Answer

So if you have this wire that&#x27;s 2019 centimetres long, you&#x27;re gonna make this rectangle with this wired that bit imager is going to be 27 years as well. So what part A. We&#x27;re gonna use that fact that the Redeemer ISI quartile two lengths plus two with and the one you don&#x27;t find the highest interest off the length Meaning solved or w So I&#x27;m gonna subtract toe L from here. Something the have to w is equal to 20 minus, shall L? My last step is to the great way to so then I know that Does your equals 10 minus l No, for part B. Me is a different color. You&#x27;re talking about the area of this rectangle. So the area this lame times with but they want you to use insurance off l. So I know that l is now and w is 10 minus l. So instead of w, I&#x27;m gonna use this w right here, which is the same Just going 10 mean this hell. So this is gonna make my area a course 10 l minus. L squared again. All I did was distribute when this route This so for part C. It&#x27;s asking you to complete this square so that you can rewrite it in terms off birth. Six form again. So to complete the square what part? C Maybe first we arrange it so it&#x27;s gonna be naked. It l squared, Minus said, Oh, so again when you&#x27;re your age and make sure that you keep the science here was about to make a mistake. So to complete the square, I say take away. I factor out the negative So it&#x27;s gonna be del square minus sent L So when you factor out the negative, you make peace to the opposites of this was negative. No, it&#x27;s positive this was positive. Now it&#x27;s negative. My next step is to then take my middle term related by two. Then divide it took you 35 and then squaring it, which is 25. So I&#x27;m gonna add 25. But I&#x27;m also a nice attract 25 because I&#x27;m gonna need that complete this queer for this part gonna have a was negative. I mean, again, this is going to compel. Plus, why squared And this princes I so need minus Johnny find here so no to rewrite the invert six form. I don&#x27;t want a double rent disease, so I&#x27;m just gonna have negative now. Plus y squared And I&#x27;m gonna distribute to Snicket it. We distributed toe this whole segment here. I&#x27;m gonna distribute it. So all of this part, it&#x27;s going to close 25. So that was part scene. The four party is word of dimensions. The process of direct and goes with the maximum area. So the maximum area is your top. So your maximum area is actually this part which is 25. So the maximum that you could get that it&#x27;s there. It&#x27;s fight because this is or part d your base use just analyzing the Vertex. It&#x27;s negative. Five. No, this is 25 actually, no, this was supposed to get here from the beginning, So this is positive. Five. So the maximum area that&#x27;s like this 25 it&#x27;s gonna be my rectangle being shy. My life

Ashley High · Answer

and this problem we&#x27;re given that this rectangle has a perimeter of 28 centimeters. So for defined the perimeter will you two times the length plus two times the width. And I will give me 28 for this rectangle. We also know that the perimeter of each triangle is 24. So each triangle actually has a link in a with of the rectangle. So one length plus one with plus the diagonal is going to equal 24 and then last. It says that I love a rectangle. So the diagonal is two centimetres longer then the longer side. So we&#x27;re just gonna make the longer side our leak. So this is part A. This is our systems off three equations. Part B wants us to make this into to a system of equations of two equations with two unknowns. So what I could do here is D equals two plus l. So I can plug that in for this deal right here. So my first equation will stay the same. It&#x27;ll be two l plus two w equals 28 and my next a crazy would be l plus W plus two plus l equals 24 and I&#x27;m gonna simplify that. So I&#x27;ll have to l Plus the view equals 22. So this is one equation are here, and this would be my other equation right here. So for part C, we&#x27;re gonna write an augmented matrix, so Els are first. So we have two and two in our Els. We have to w wonder you and then are constants are 28 and 22 d is asking us to find the link of the Asari, the length of the dimensions. So when you defied l and W So our matrix, we&#x27;re going to do some operations. So first, we&#x27;re going to make this equal zero so we can do row one minus row two, and that&#x27;s gonna replace row two. So world one will stay the same. So we have to minus 20 and two minus one is one and then 28 minus 20 to 6. So you got legal, dear? Oh, the whole top row who can actually divided by two. So run divided by two equals ro one. So my new matrix will be one one 14 016 So if I look here, this bottom row has one in six will. That one is our w. So we know that one w equal six So w equal six and then the top row is our length plus one w So we know one w equal six equals 14. So So track six from both sides and the length will equal eight. So the dimensions of our rectangle are that are w equal six in our length equals eight Part E wants us to find the diagonal so we know from part ay that are diagonal is two plus l We know our l equals eight sarda agonal equals two plus eight so the length of the diagonal equals 10.

Zach Steedman · Answer

All right. So we know that the perimeter of this rectangle is gonna be to lengthen open, like capital l So it looks a little bit less like a 12 length supposed to with secrets 24 in the area would be one of times with I&#x27;m gonna take this equation here and solve for length. Wanna And so that would give me if I subtract two w It&#x27;s gonna be to length how this let&#x27;s divide everything by two. And so now we just have length Plus with equals 12. Or in other words, length would be equal to 12 minus with. And so I&#x27;m gonna use that to plug that in here and solve four w Since of length, I&#x27;m gonna have 12 minus w. And it&#x27;s gonna be multiplied times w equal to 32 we&#x27;re gonna have a squared in here. But that&#x27;s OK. And off kind of show you Why? Um, so we have a 12 times w so that&#x27;s 12 w. We have a negative w squared and we have equal to 32 and we can set this up like a quadratic equation. If we put negative w squared here less 12 w subtract 32 from both sides, leaving us with zero here. And if he&#x27;s attract 32 it looks like this equals 01 more thing. Let&#x27;s multiply everything times a negative one and rewrite this. So we got W squared and now we got a minus 12 w and a plus 32 equal to zero. And let&#x27;s go ahead and factor this. And if we factor this out, it turns out that this could be factored to be W minus eight and W minus four equal to zero. And that&#x27;s actually good, because that means that the two solutions or W, can be eight or W can be four. And that is because the width really could be kind of drawn kind of like this, so that the width is a longer one. The answer to this equation is that the length and the width are eight and four, and maybe it&#x27;s four times a day and you switch it. But if I plug in eight into this equation eight times what is 32? Well, four or if I plug in for four times eight is 32. Either way, these air the solutions eight times four is your length in your width

Problem

The sum of the lengths of the legs of a right tri…

Problem 18 Hard Difficulty

Answer

a) $12-x$
b) $y=x(12-x)$
c) See graph
d) 6 ft by 6 $\mathrm{ft}$

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Ann Xavier Gantert

Algebra 2 and Trigonometry

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

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a) $12-x$b) $y=x(12-x)$c) See graphd) 6 ft by 6 $\mathrm{ft}$

Algebra 2 and Trigonometry

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Grace H.

Anna Marie V.

Heather Z.

Kayleah T.

a) $12-x$
b) $y=x(12-x)$
c) See graph
d) 6 ft by 6 $\mathrm{ft}$

Ann Xavier Gantert

Algebra 2 and Trigonometry