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Amanda drove 40 miles. Then she increased her rate of speed by 10 miles per hour and drove another 40 miles to reach her destination. If the trip took 1$\frac{4}{5}$ hours, at what rate did Amanda drive?

$40 \mathrm{mph}, 50 \mathrm{mph}$

Algebra

Chapter 2

THE RATIONAL NUMBERS

Section 7

Solving Rational Equations

Fractions and Mixed Numbers

Decimals

Equations and Inequalities

Oregon State University

Harvey Mudd College

Baylor University

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So Amanda travels for 40 miles at some unknown speed. She then travels another 40 miles, and this time her speed, where her rate is 10 miles an hour faster than she was going for the 1st 40 miles. And we know the entire trip takes one and 4/5 hours. Well, one in 4/5 is the same thing is saying 1.8 hours, and we have to find out how fast she was going on each segment of her trip. And in order to do that, we're going to create a rational equation. So the first thing you must keep in mind is that time plus time will always equal time, and the second piece of information that you have to recall is the fact that any time you take a rate and multiply it by, it's time you will get its distance. For example, I'm traveling 60 MPH for two hours, and I will have traveled 120 miles. So this way you can see that rate times time equals distance connection. So now, since we are more worried about time, what we're gonna do is we're going to take that rate times time equals distance and we're going to solve for time. So if I were to take the rate times time equals distance and sauce or divide both sides by rate, then I know that time can be found by taking a distance divided by a rate. So when I look at the first leg of Amanda's journey, if I take the distance 40 miles and divide it by her rate, I will get the time it took her to do that first leg. And then if I do the same thing for her second leg of the journey, if I take her distance and I divided by her rate, I will get the time it took. And I know that when I put those two parts of the journey together, it took 1.8 hours. So we need to solve this rational equation to find out Amanda's speed for the first leg of her journey and the second so, in order to do that, keep in mind that we are always allowed to multiply both sides of an equation by the same non zero number, and it will remain in balance or it will remain equal. So I'm going to multiply both sides of the equations by the two denominators. So I'm going to multiply this side by an X Times an X Plus 10, and I'm going to multiply the other side by an X Times X plus 10. So what do I have on the right side? I'll have 1.8 x Times X plus 10 on the other side. I'm going to have to distribute. I'm going to have to distribute this quantity to both fractions. And in doing so, I will end up with 40 over X Times X Times X plus 10 and then for the second fraction I'll have 40 over X plus 10 times X Times X plus 10. And what will happen now is when I multiply this first, um, term. The X from the denominator will cancel with the X from the numerator and I'll be left with 40 times the quantity of X plus 10. And then I'm going to add to that what I get when I multiply here and when I multiply here, the X plus 10 in the denominator will cancel with the X plus 10, and I'm left with just 40 x. And if I distribute on the other side of the equal sign, I get 1.8 x squared plus 18 x. So now we're no longer dealing with a rational equation. We're now dealing with a quadratic equation, so I'm going to have to distribute to clean out the parentheses, so I get 40 X plus 400 plus 40 x equals 1.8 x squared plus 18 X. And the easiest way to solve the quadratic equation is to get all variables and terms on one side of the equal sign and get it equal to zero, which we refer to as being in standard form. So I am going to subtract 40 x on both sides. I'm going to subtract 40 x again on both sides, and I'm going to subtract 400 from both sides. And in doing so, I get a new equation. Zero equals 1.8 x squared minus 62 x minus 400. I still have to solve that quadratic equation, and I could do one more thing to make it a little easier to solve again. I'm allowed to multiply both sides of an equation by a none zero number and still maintain the balance or the equality. I'm going to multiply both sides by 10 because I don't like working with decimals. So on the left I'll have zero. And when I distribute that 10 to this 1.8 coefficient, I'll now have 18 x squared. I'll have to distribute here minus 620 and I'll distribute to here and I'll get 4000. I noticed that all of the coefficients are even so. Therefore, everything is divisible by two. So if I were to divide both sides of the equal sign by two, I'm going to get an easier quadratic equation that I need to solve. So I'm going to distribute that to into all three terms of the numerator, and I will get nine X squared minus 310 x minus 2000 equals zero. Now, to solve the quadratic equation that is of this style, it's probably best if we apply the quadratic formula. And the quadratic formula says X equals negative B plus or minus. The square root of B squared minus four a. C all over two A and a is the coefficient of the quadratic term B is the coefficient of the linear term, see is the constant when our equation is in standard form. So in our problem we have a being nine. B is negative 310 and C is negative 2000. So we're going to substitute those values into the quadratic formula. So are unknown. X Value, which represents the rate of Amanda's first leg of the trip, will be the opposite of B, which is now positive. 310 plus or minus B squared, which would be negative 310 squared minus four multiplied by a and then multiplied by C. And then it's all over two times a and I'm going to do some clean up. And when I clean up, excuse me. Everything underneath the radical I'm going to get X equals 310 plus or minus. The square root of 16 are actually 168,100 and then it's all over 18. And if I were to take the square root of 16 one of try again 168,100 I will end up with 410 now. Technically, there's two answers here. We can follow the top path and add the two in the numerator or I can follow the bottom path except the bottom path leads us to a negative answer. So we are going to eliminate or throw away that answer because keep in mind what X waas again X was a rate or a speed that Amanda was traveling, so she can't be traveling at a negative speed. So therefore the only viable answer would be following the positive path, and we will get an answer of 40. So we now know that X is equal to 40. So let's travel back up to the original set up. So Amanda was going 40 MPH on the first leg of her journey. And then, if we do 40 plus 10, she was traveling 50 MPH on the second leg of her journey. So there are two parts to the answer. She was originally going 40 miles an hour, and she finished her trip traveling 50 MPH

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