in-43-47-express-each-answer-in-simplest-radical-form-the-dimensions-of-a-rectangle-are-sqrt250-mete

Question

In $43-47,$ express each answer in simplest radical form.
The dimensions of a rectangle are $\sqrt{250}$ meters and $\sqrt{1,440}$ meters.
a. Express the perimeter of the rectangle in simplest radical form.
b. Express the length of the diagonal of the rectangle in simplest radical form.

Julie Silva · Accepted Answer

So in this problem we were given the dimensions of a rectangle. It&#x27;s to square the 250 m by the square with 1400 for the beaters and import A We want to find the perimeter will remember the formula to find the permanence of the rectangle is two times toe with plus two times the length. So in this case, we have our within link. So we just need to substitute them into our formula. So love the perimeter equals two times the square of the 250 plus two times the square root of 1440. Well, in order to rad radicals, we need to simplify Nimble first. So we&#x27;ll start with our first term. We have to square with a 250. I&#x27;m gonna bring down the coefficient of to Now the perfect square 25 goes into it 10 times, so we&#x27;ll rewrite it as to square the 25 Times Square. The 10 now for the second term again, I&#x27;ll bring down the to and the perfect square 144 goes into it 10 times. So we&#x27;ll rewrite this as two square with the 144 times the square of the time. All right, so again bring down the to We have the square of the 25 which is five, and we can simplify to square the 10. We&#x27;ll bring that down for the second term again. We&#x27;ll bring down the to the square with a 140 four&#x27;s 12th and we can&#x27;t simplify the square of the time. Now let&#x27;s multiply those coefficients. We first step two times five, which is 10. So we have 10 Route 10 plus two times 12 which is 24. So we have 24 Route 10. I see that these air now both lake radicals because they have the same radic. An. So we combine the coefficients while 10 plus 24 is 34. So we&#x27;ll have 34 Route 10 and our units in this problem is meters. So now we found the perimeter. It&#x27;s 44 times the square, the 10 m. All right, so now let&#x27;s go back to part B. We want to find the length of the diagonal. Well, what&#x27;s picture? What this would look like here with the my rectangle. So if I put in my length. The square of the 144 sorry square with 1444 and our with two square that you 50. We want to find a diagonal will remember in the diet of rectangle each of our angles of right angles. So this means that this bottom right part is a triangle and we&#x27;re trying to find the pirate party. We&#x27;re trying to find the iPod news. So to do this, we&#x27;re going to use the boat staggering Terram, which remember states a squared plus B squared is equal to see square. All right, well, A and B are legs. So in this case, it&#x27;s us to sidelines. So we&#x27;re gonna have to square root a 1440 that&#x27;s gonna get squared. Plus, we have to square the square root of 250 and it will be equal to our Aponte new squared. So x square. All right, Well, remember, whenever we square a square root, it essentially cancels it out. So when we square the square with 1400 for the were simply left with just 1440 and similarly, when we square the square of the 250. We&#x27;re just going to get 250. And this is all equal to X squared. Well, now we can add We have 1440 plus 250 which is equal to 1690. It&#x27;s equal to X squared and then to solve for X, we have to take the square of the both sides. Now square the vex square. This just tax, which is great. But now we have to simplify the square root of 1690. Well, the perfect square 169 goes into it 10 times. So we&#x27;ll rewrite this as the square with a 1 69 times square. The 10 Well, the square, the 1 69 is 13 and we can&#x27;t simple five square. The tents will bring it down. And now we found the length of the diet Agdal. And don&#x27;t forget units. It&#x27;s meters, so the length of our diagonal will be 13 times square with the 10 m

Julie Silva · Answer

So in this problem, we&#x27;re told that in a right triangle we have two legs that are have length of 18 square that 18 centimeters and the square of the 32 centimeters now in part A. They want us to find the high Ponte news. Well, let&#x27;s think about what this triangle would look like. So we would have a right triangle. And because the two legs of the squares of 18 that would be these two sides and we&#x27;re looking for the taipan news. So to do this, we&#x27;re going to use the Pythagorean theorem, which, remember the Pythagorean theorem is a squared plus B squared equals C squared, where A and B are the legs and seized iPod news. So let&#x27;s substitute these values into our formula so we would have the square root of 18 getting squared, plus the square of the 32 being squared. And it&#x27;s equal to our high pot news, which is X word. All right, well, now we have to square the square of 18 while when we square a square root, they essentially cancel each other row. So the squares of 18 squared is simply just 18 and the same is true when we square the square with a 32. We&#x27;re just left with 32 in the end is equal to x word. Well, now we have 18 plus 32 which is 50. We have 50 is equal to x word. So to solve for X we&#x27;re going to take the square with the both sides Because the square the vexed where this just X Then we have to square with a 50 but it can be simplified. The perfect square 25 goes into it two times. So we&#x27;ll rewrite it as the square of the 25 times the square the to and the square The 25 is five. So we&#x27;re left with five route to So now we&#x27;ve found our high pot news. It will be five route to will throw with our units here. Centimeters. Okay, great. Now for part B. We want to find the perimeter Will remember to find the perimeter. We need to add up all the sidelines. So our side lanes are the squares of 18 plus two square the 32 plus our high ponte news which remember, we just found to be five route to so What we need to do is add these links together. But remember, before you can add radicals, you have to simplify first. So let&#x27;s start with the square of 18. Well, the perfect where nine goes into 18 2 times. So we&#x27;ll rewrite this as to square the nine Times Square the to now for the square, the 30 to the perfect square 16 goes into it two times, so we&#x27;ll rewrite this as to square the 16 times Square the to, and we can&#x27;t simplify five routes, so we&#x27;ll bring that down. All right, so from our first term to square, the nine is three. We&#x27;ll have three brute to from the second term, the square. The 16 is four. It&#x27;s a while. Four route to and again, we&#x27;ll bring down our last year. Now all of our radicals air simplified and each of these terms of like terms because they all have the same radic in. So we just need to combine the coefficients well. Three plus four is seven and seven, plus five is 12 so we&#x27;ll have 12 fruit, too, and we&#x27;ll put in our units here, which is centimeters, So the perimeter of this triangle is 12 or two centimeters

Finian Lickona · Answer

all right, so in this exercise, we are given that the length of one of the legs of a right I saw sleaze triangle is route 98 inches by means that the length of the other egg is also route 98 inches. The first portion of the question asks us to find the high pot news with second portion asks us to find the perimeter. So we&#x27;re gonna start by finding the high pod news. Now, if you&#x27;ll recall this Ah, iPod news value is given by the Pythagorean Fear. Um, a squared plus B squared equals C squared where a and fear the lengths of the lakes. So a squared would be the squared of 98 squared, which is just going to be 98. And that&#x27;s going to also apply to the second leg. So we see that the high pot news squared length is 196. Now, if we want the actual high pot news link, we need to take the square root of both sides. Fortunately, 196 is a perfect square. It is the square of 14 so the length of the high pot news is 14 inches. Now, for the second part of the problem, we have to find the perimeter so that simply amounts to adding the quantities. 98 Route 98 plus Route 98 plus 14. Now let&#x27;s see if we can simplify these radicals. A bit 98 is written. Can be decomposed into two times. 49 49 is a perfect square. We can factor it out as a seven, so we have a seven route to plus seven route, too, plus 14. Now we can combine the route to terms into 14 route, too, but you can&#x27;t add a non radical terms with radical terms. However, we can factor 14 out of both of these terms and simply rewrite this as 14 times one plus route, too. So are high pot. News length is 14 and our perimeter length is 14 times the quantity one plus route, too

Finian Lickona · Answer

all right and this exercise were given a I saw sleaze triangle, and we&#x27;re told that the length of each of the longer sides is 500 feet, and the length of the third side is 45 feet. And the question asks us to find the perimeter of the triangles. That means that we have to add radical 500 plus a radical 500 plus a radical 45. Now, in order to do this, we&#x27;re gonna have to simplify them also that the radical terms are matching. Now I see that 500 is five times 100. 100 is a perfect square. It factors out of the radical is 10. So that becomes 10. Route five. Same thing with this term right here and 45 can be written as nine times 59 is a perfect square. It factors at his three, so this expression becomes three refire now that all the radical terms match I can add their coefficients. 10 plus 10 plus three is 23 Route five. So our final answer is that the perimeter is 23 ri five feet

Julie Silva · Answer

So in this problem, we&#x27;re told that the three sidelines for triangle are the square, the 75 the square, the 27 in the square root of 108. And we want to find the perimeter. Will remember to find the perimeter. We simply add up all the sidelines. So we need to add to square the 75 plus the square, the 27 plus the square root of 108. Okay, great. Now we just need to simplify. So let&#x27;s start by simplifying each of our radicals. So you know, the permanent equals Well, we&#x27;ll start with the square that 75 the perfect square 25 goes into it three times. So we&#x27;ll rewrite this as to square the 25 Times Square. The three. Now for our second radical to square the 27 the perfect square nine goes into it three times. So we&#x27;ll rewrite this as two square than nine times to square the three. And then lastly, we have the square root of 108. Well, the perfect square. Um, I got lost there for a second there. Ah, 36 goes into it three times, so we can rewrite this as to square the 36 times the square, the three. Now let&#x27;s go ahead and simplify each of these. Well, the square the 25 is five. So we&#x27;ll have five or three for a next term. The square of the nine is three. So we&#x27;re left with three for three. And from the last term, the square of the 36 is six. So we have 633 All right, great. Now we have, like, two terms here because they all have the same radic. And so we just need to simplify by combining the coefficients well. Five plus three is eight in a plus. Six is 14. So the perimeter will be 18. Brew three now, remember, they leave us units so the units for our per mother will be inches. So our final answer is that the perimeter of this triangle will be 18 route three inches

Foster Wisusik · Answer

Okay, so this question says we have a rectangle and of course it is a length and a with, but it actually tells us that there&#x27;s a relationship between the length and the width. So it says that the length is four feet longer than twice the with, so the length is equal to two W plus four, and it also says the area is 38. So from here, we can actually solve for the with because we know the area for a rectangle is just length. Times with so 38 is equal to to W plus four times W or to W squared plus four w equals 38 or to W squared plus four W minus 38 equals zero. So now we can solve this quadratic. So we&#x27;re going to complete this square by transforming this into Vertex form. And we&#x27;ll do this by finding D, which is be over to A and E, which is C minus. B squared over for a for our rectangle. So says that D is equal to be over to a and that&#x27;s just four divided by two times two So D is just one, and then e is C minus B squared over for a So we get negative. 38 minus 16 divided by 16. We&#x27;re sorry. 16 to vote by A rather which gives us negative 40. So now we can write quadratic as two times w plus one minus 40 or sorry. The w post one squared minus 40 equals zero for W plus one squared is equal to 20. So this means that W plus one is equal to plus or minus the square root of 20. And again I added forgettable sides and divided by two to isolate the w plus one squared. So now we know that w plus one equals 20 for a W equals W plus one equals minus 20. So w is equal to route 20 minus one or w equals negative 20 minus one. And we can&#x27;t have two different withs source one to its pick. Well, negative 20 negative squared of 20 is going to be like negative 4.5 or somewhere around there somewhere between negative for negative five. So we have a negative minus and negative on this right side, and withs cannot be negative. So we&#x27;re gonna throw this solution out so this W is the one that we take. So we know that the with is equal to squared of 20 minus one. Or if we want to simplify our radicals, that&#x27;s squared off five times for minus one or w equals to root five minus one feet. And now it wants us to also find the length. So you know that length is equal to going all the way back to W plus four. So L is equal to two times to root five minus one plus four, which means that l is equal to four root five minus two plus four. So plus two. So now we have the dimensions for our length and or with, and it wants us to check that this gives us the area. So area is equal the length times with so the length is four root five plus two, and with is to root five minus one. So we&#x27;re gonna have to foil this out. So first will distribute this term to get eight times five. Then we get minus for rue five. Then we get plus two, Route five minus two. Sorry, plus for five. And these two terms cancels or just given a times five just 40 minus two, which is 38 square feet. And that is what they gave us for the original area. And then it wants approximations for each of these. So l is approximately 10.94 and W is approximately with 3.47 and now we&#x27;re done.

Problem 47 Hard Difficulty

In $43-47,$ express each answer in simplest radical form.
The dimensions of a rectangle are $\sqrt{250}$ meters and $\sqrt{1,440}$ meters.
a. Express the perimeter of the rectangle in simplest radical form.
b. Express the length of the diagonal of the rectangle in simplest radical form.

Answer

a) 34$\sqrt{10} m$
b) 13$\sqrt{10} m$

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a) 34$\sqrt{10} m$b) 13$\sqrt{10} m$

Algebra 2 and Trigonometry

Ann Xavier GantertAlgebra 2 and Trigonometry

a) 34$\sqrt{10} m$
b) 13$\sqrt{10} m$

Ann Xavier Gantert

Algebra 2 and Trigonometry