<p>Could anybody please explain to me how do to #'s 17 and 18 on p. 412?</p>
<p>Let x = distance to work.</p>
<p>(x/45mph) + (x/30mph) = 1 hr (total driving time of morning + evening)
Solve for x by converting the above eq. to the LCD form:</p>
<p>2x/90 + 3x/90 = 5x/90 = 1 hr
x = 18 miles</p>
<h1>17:</h1>
<p>Plug in (0,p) and (5,t) into the equation. You will get:</p>
<p>0 = p^2 - 4
p^2 = 4
p = 2 or -2</p>
<p>and</p>
<p>5 = t^2 - 4
t^2 = 9
t = 3 or -3</p>
<p>Therefore, the possible points are (0,2), (0,-2), (5,3), and (5,-3). To maximize the slope, you will want to maximize the y difference, so you choose (0,-2) and (5,3) as your two points:</p>
<p>slope = 3 - (-2)/5 = 1</p>
<h1>18:</h1>
<p>Algebraic solution:</p>
<p>rate x time = distance</p>
<p>So for the first trip:</p>
<p>45*t = d, where t is the time for the first trip and d is the distance travelling one way on the route</p>
<p>For the second trip:</p>
<p>30*(1-t) = d, since the time for the second trip plus the time for the first trip (t) must add to 1 hour.</p>
<p>Equate the two to get</p>
<p>45<em>t = 30</em>(1-t)</p>
<p>Solve to get t = 0.4 hour</p>
<p>Putting this back into the first equation, you obtain</p>
<p>45*(0.4) = 18</p>
<p>So the final answer is 18.</p>
<p>Or</p>
<p>You can use the shortcut for finding the total rate from two rates for trips of equal distance:</p>
<p>R = 2<em>rate1</em>rate2/(rate1 + rate2)</p>
<p>Putting the values in, you will get</p>
<p>R = 36</p>
<p>So the total rate is 36 miles per hour. Esther drove for an hour total, so she must have driven 36 miles in total. The question is asking for the miles of the drive to work, however, so you divide this distance by 2 and obtain 18 miles, the same as before. :)</p>
<p>A final method, not the most efficient, but a method nonetheless, would be to plug in "answers" and verifying each guess. Of course, we have no answers on grid-ins, but we can make them up by estimating:</p>
<p>Estimate what the answer might be. The first step would be to do a straight average of the two rates (which you KNOW will not be correct, since that is the Joe Bloggs approach): 37.5 miles per hour. You know that the true total rate will be less than that because Esther spends more time driving at the lower rate. So you can guess maybe 37 miles per hour, giving you a distance of 37 miles, first.</p>
<p>So the distance per trip would be 18.5 miles. This is probably not the right distance, since the question will probably want to use a nice integer, but you try it anyways.</p>
<p>You get the times for the two trips to be 37/90 hour (an ugly number!) and 37/60 hour. These add up to 37/36 hours, a little too much. So you try a lower distance, 36 miles.</p>
<p>The distance per trip would be 18 miles. And the two times would be 0.4 hour and 0.6 hour, and they add up to 1 hour. Bingo! 18 miles is your answer. :)</p>
<p>This problem is easier to see when you graph it out.
x = y^2 -4 is a sideway parabola with opening facing to the right.</p>
<p>Bec. you want to find the "greatest possible slope" of line l,
you can assume its slope m > 0.
Now draw line l with a positive slope intersecting the parabola at
(0. p), and (5, t).</p>
<p>(0, p) == a y-intercept for the parabola.
Therefore plug x = 0 into its eq. to solve for p:
0 = p^2 -4
0 = (p-2) (p+2), p = 2, or p = -2.
Bec line l has positive slope, it must intersect the parabola at the lower of the 2 y-intercepts, thererfore p = -2. </p>
<p>(5, t) is a point on the parabola so solve for t by plugging in x = 5:
5 = t^2-4, t = 3 or t = -3.
Again bec line l has slope > 0, t = 3.</p>
<p>Solve for slope m using the 2 points (0, -2) & (5, 3):
m = 0 - 5 / -2 - 3 = 1</p>
<h1>17:</h1>
<p>The easiest thing to do is to draw this. Keep in mind that the equation given to you is defined in terms of y, so it's a sideways parabola. When you draw it you should realize that the line that keeps give the "greatest" (or most positive) slope is going to be one in which you have the smallest value possible on the left (for x = 0) and the greatest value possible on the right (for x = 5). You can put the equation given in terms of x in order to solve for these values,</p>
<p>y = sq.rt(x+4) and y = -sq.rt(x+4)</p>
<p>where the positive square root is the upper part of the parabola and the negative square root is the lower part of the parabola. Solving gives you that p should equal -2 and t should equal 3. Now just use the slope formula to find the slope for your two points:</p>
<p>(0, -2), (5, 3)</p>
<p>m = [3-(-2)]/(5-0) = 5/5 = 1</p>
<h1>18:</h1>
<p>rate = distance/time, so time = distance/rate</p>
<p>x = distance to work (or distance from work to home, since they're the same)</p>
<p>x/45 + x/30 = 1 (rate/distance + rate/distance = total time)</p>
<p>Now just solve for x:</p>
<p>2x/90 + 3x/90 = 1
5x = 90
x = 18 miles</p>
<p>Thanks for the explanations.</p>