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One base of a trapezoid is 8 feet longer than the other base and the height of the trapezoid is equal to the length of the shorter base. The area of the trapezoid is 20 square feet.a. Find the lengths of the bases and of the height of the trapezoid in simplest radical form.b. Show the lengths of the basezoid is equal to one-half the height times the sum of the lengths of the bases.c. Write, to the nearest tenth, rational approximations for the lengths of the bases and for the height of the trapezoid.

a) $-2+2 \sqrt{6}, 6+2 \sqrt{6},-2+2 \sqrt{6}$b) See derivationc) $2.90,10.90,2.90$

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 1

Real Roots of a Quadratic Equation

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

Baylor University

University of Michigan - Ann Arbor

Idaho State University

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Okay, This question asks us about a trap. Is oId where it says one base has a length and we'll call that be, and then the longer base has the length. B plus eight in the height also happens to have a length of B, and the area is equal to 20. So now we can find our value for these bases and the height by using the fact that area of a trap is laid is just a chew over two times B one plus B two or, in our case, H is equal to be B one is being, and B two is B plus eight or a equals B over two times to be plus a or a equals be times B plus for or area is equal Toby squared plus four p. But we also know that are areas 20. So we're left with 20 equals B squared, plus four B or B squared plus four B plus 20 equals zero, and this is a quadratic equation that we can solve to get our base. So to do this will complete the square by turning a quadratic into its vertex form. And I I changed the letters inside this equation Hair just so you don't get confused about what be we're talking about. And we said that D is equal to rxa coefficient divided by two times are quadratic coefficient and then e is equal to C minus. Rxa coefficients squared over four times our X squared coefficient. So now we can plug into our equation to solve for these, so we said, are quadratic is B squared plus four B plus 20 equal zero so de would be equal to for over two or two. And then the E would be equal to 20 minus 16/4 or 20 minus four, which is 16. So now we can read our equation as B plus two quantity squared is equal to 16 or this is a sorry. We made a sign air here. This is minus 20. So this should be minus 24. It's not actually gonna work out that nice. So now that this is all corrected, we see that B plus two squared is equal to 24 so B plus two is either equal to positive 24 or negative through 24. And again, I apologize for my initial algebra air or copying era. Rather, we'll fix that over here, too. So we have our solutions as B is either Route 24 minus two or negative Route 24 minus two, and this solution gets thrown out because it's negative and lengths can't be negative. So we're just left with B equals squared of 24 minus two or B equals square root of 24 6 times for so we can get our final form for B, which is to route six minus two feet and will use this in all our other calculations because they all are related to be so. Our area is equal to our expression over here B squared plus four B, and that's also equal to our height, divided by two plus the sum of our basis. So we said that be was equal to to Route six minus two, so we'll spoil these out and see if we get original area. So for our first term, we get to route six times to Route six, which is four times six or 24. Then we get minus to we get minus two times two times two times or six times or six so and then our last term is plus four and then we're also adding four times. Or sorry, this is just a Route six. This is a Route six in here, plus four times to Route six minus two or 24 minus eight. Route six plus four plus eight. Route six minus eight, which is equal to 28 minus a or 20 square feet. So this does indeed work out. So we did find the right value. So now it wants us to approximate the base, the height and the other base. So we said, the short base is just equal to coming over here to Route six minus two, and H is also equal to 26 minus two. But the other base we'll call Beat Long is equal to the short base plus eight. So to route six plus six and finding approximations for each of these we get 2.9, 10.9, 2.9 for the shorter base and 10.9 for the longer base, and these are our final lengths

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