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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.

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

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Chapter 3

REAL NUMBERS AND RADICALS

Section 4

Adding and Subtracting Radicals

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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So in this problem we were given the dimensions of a rectangle. It'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'll start with our first term. We have to square with a 250. I'm gonna bring down the coefficient of to Now the perfect square 25 goes into it 10 times, so we'll rewrite it as to square the 25 Times Square. The 10 now for the second term again, I'll bring down the to and the perfect square 144 goes into it 10 times. So we'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'll bring that down for the second term again. We'll bring down the to the square with a 140 four's 12th and we can't simplify the square of the time. Now let'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'll have 34 Route 10 and our units in this problem is meters. So now we found the perimeter. It's 44 times the square, the 10 m. All right, so now let's go back to part B. We want to find the length of the diagonal. Well, what'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're trying to find the pirate party. We're trying to find the iPod news. So to do this, we'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's us to sidelines. So we're gonna have to square root a 1440 that'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'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'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'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't simple five square. The tents will bring it down. And now we found the length of the diet Agdal. And don't forget units. It's meters, so the length of our diagonal will be 13 times square with the 10 m

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