Hot X: Algebra Exposed

Transcription

1 Hot X: Algebra Exposed Solution Guide for Chapter 3 Here are the solutions for the Doing the Math exercises in Hot X: Algebra Exposed! DTM from p Our units aren t consistent, and since the answer should be in miles/hour, let s convert the minutes to hours: 2 minutes = 2 min hour 60 min = 2 60 hour = 5 hour and 8 minutes = 8 min hour 60 min = 8 60 hour = 3 0 hour Let s call Jessica s rate in still water r. With a river current speed of 2 miles/hour, that means her rate downstream (will be faster) is r + 2, and her rate upstream (will be slower) is r 2. This is a round trip problem, so let s set up our r*t = d chart; one row for the downstream trip and one row for the return upstream trip, and fill in what we know: downstream trip r upstream trip r And we know that we can multiply across to get expressions for the distance Jessica

2 traveled in each case, so let s fill that in, too: downstream trip r upstream trip r (r + 2) 5 3 (r 2) 0 And since (typical of roundtrip problems), the downstream distance is the same as the upstream distance, we can set these two distances equal to each other and solve for r! in other words, we know this is a true statement: 3 (r + 2) = (r 2) 5 0 To make this easier to deal with, let s multiply both sides by 0 and get rid of those pesky fractions: 0 3 (r + 2) = 0 (r 2) 2(r + 2) = 3(r 2) 2r + 4 = 3r 6 0 = r 5 0 So Jessica s rate is 0 miles/hour. And that s what the problem asked for. Done! Answer: 0 miles/hour 3. Looks like our units are consistent, so let s go ahead and see what we can fill in for the chart. This is a moment in time problem, so we know the time traveled will be the same in both rows; we ll just call it t, which will be the moment when they are 50 miles apart. We know their rates, 30 and 70, and then we just multiply across to fill in the final distance column. First train 30 t 30t Second train 70 t 70t

3 Now, when the two trains are 50 miles apart, that means that the distance the first train has traveled, plus the distance the second train has traveled, adds up to be 50 miles. Make sense? If it helps, check out the picture below: This means that 30t + 70t = 50. Great! Let s solve it: 30t + 70t = 50 00t = 50 t = t = 2 hour In other words, a half hour. And that s how long until they are 50 miles apart! Answer: 2 hour 4. This is a round trip problem, so let s make the first row the upstream trip and the second row will be the downstream trip. Look at our units; for rate we have km/hour, and so that s what our rate answer should be in, and for time, we have minutes. Let s convert our minutes to hours, and we get: 5 minutes = 5 min hour 60 min = 5 60 hour = 4 hour (time it takes to go upstream) and 3 minutes = 3 min hour 60 min = 3 60 hour = hour (time it takes to go downstream) Let s call the speed of the current c. That means that Amanda can travel (3 c) km/hour upstream, and (3 + c) downstream. Let s fill in what we know on the chart, all in km and hours. Don t be afraid of the fractions! We ll get rid of em soon enough.

4 upstream (3 c) 4 downstream (3 + c) (3 c) 4 (3 + c) And since the distance upstream is the same as the distance downstream, we can just set these two last expressions equal to each other, and solve for c! 4 (3 c) = (3 + c) First off, let s multiply both sides by to get rid of the fractions: 4 (3 c) = (3 + c) 5(3 c) = (3 + c) 5 5c = 3 + c 2 = 6c c = 2 So the current s speed is 2km/hour. Done! Answer: 2 km/hour 5. Looks like all our units are in miles and hours. Nothing to convert, great! Okay, this is a moment in time problem, so we ll call that moment t, and let s fill in what we know: Crystal 600 t 600t Dina 650 t 650t And we know that at the moment in time when the planes pass each other, the distance Crystal s plane has traveled, plus the distance Dina s plane has traveled, adds up to equal 2500 miles.

5 So that means this is a true statement: 600t + 650t = And now we just solve for t! 600t + 650t = t = 2500 t = t = 2. After 2 hours, their planes will cross and they can wave to each other! But wait that s not what the problem asked for. The problem wanted to know how far from New York Crystal will be when they pass so, after 2 hours, how far has Crystal traveled at a speed of 600 miles/hour? That s just a simple application of the r*t = d motion formula! r*t = d 600(2) = d d = 0 miles. Done! Answer: 0 miles from New York 6. This is just like the example on p.88-90, so check that out for a diagram of these kinds of problems, and to get a sense of what happens when someone laps someone else. Okay, first off, we have meters and hours, so the units are consistent. Great! This is a moment in time problem. So the time traveled for both rows will simply be t, which stands for the moment we re interested in: the moment when Devon laps Erin. Since the track is 50 meters around, that means at the moment of lapping, Devon will have traveled 50 meters more than Erin. Make sense? So, at our special time t, if Erin s distance is d, then Devon s distance will be d We also know that Devon s speed is 50 meters/hour faster than Erin s. So if Erin s rate of speed is r, then Devon s speed is r Let s fill this stuff in!

6 Erin r t d Devon r + 50 t d + 50 Hm, two equations but three unknown variables? Let s just see what happens when we do a little substitution. You ll see that in these lapping problems, one of the variables usually disappears pretty easily! Remember, we want to find t for our final answer. So the above chart means the following is true: rt = d (r + 50)t = d + 50 Using the first equation, let s substitute rt for d in the second equation, and we get: (r + 50)t = d + 50 (r + 50)t = rt + 50 Now let s simplify: (r + 50)t = rt + 50 rt + 50t = rt + 50 And hey look at that! The rt terms both subtract away, and we re left with: 50t = 50 t = t = 3 So after 3 hours, Devon will lap Erin. Done! Answer: 3 hours