<p>17) In the xy coordinate plane, the graph of x = (y squared) - 4 intersects line L at (0,p) and (5, t). What is the greatest possible value of the slope of L?</p>
<p>18) Esther drove to work in the morning at an average speed of 45 miles per hour. She returned home in the evening along the same route and average 30 miles per hour. If Esther spent a total of one hour commuting to and from work, how many miles did Esther drive to work in the morning? </p>
<p>Thanks!</p>
<p>For the first problem, substitute the given values of x (0 and 5) into the equation, and solve for y. for 0, y=2,-2, and for x=5, y=3, -3. the slope formula is change in y over change in x. for the greatest slope, we want the change in y to be the most, thus lets pick -2 and 3, the change being 5. so 5/(5-0)=1, therefore, 1 is the greatest slope the line can have. </p>
<p>For the second problem, lets say that the time she took to go to work is x, and the time it took to come back is y. Therefore, x+y=1, since the total time is one hour. Now, since d=vt (distance=velocity*time), t=d/v, and we know the distance is the same in both cases. So lets substitute, d/45+d/30=1, multiply both sides by 90, 5d=90, and thus d=18 miles. (at least I think so, tell me if that’s not the right answer.)</p>
<p><a href=“http://talk.collegeconfidential.com/sat-preparation/441436-bb-page-412-17-a.html[/url]”>http://talk.collegeconfidential.com/sat-preparation/441436-bb-page-412-17-a.html</a></p>
<p>for number 1:</p>
<p>plug both coordinates into the x=y^2 - 4 equation</p>
<p>so 0= P^2 - 4
p^2 =4
p= -2 or 2</p>
<p>5= t^2 -4
9=t^2
t= -3 or 3</p>
<p>to find the greatest possible slope, figure out which combination of the possibile y coordinates creates the greatest sum or difference (it doesn’t matter which order the numbers come in because the difference will be made up for in the x1-x2 vs x2-x1 denominator choice anyway)</p>
<p>if p = 2 and t = -3, then slope = 5/-5 =-1
p= -2 and t= 3, then the slope is (3–22)/(5-0) = 1 i think this is the greatest but i’ll check the other 2 possibilities
p= 2 and t=3, then (3-2)/(5-0) = 1/5
p=-2 and t=-3, then (-2–3)/-5 = -1/5</p>
<p>so the greatest possible slope is 1</p>
<p>
OK i set up the following 2 equations, assuming x is the distance in miles between home and work</p>
<p>Xmi times 1hr/45mi (i inverted the mph, to cancel the units. I can’t remember if I learned this in any math class but it’s useful in chem/physics)</p>
<p>and Xmi times 1hr/30mi</p>
<p>the mi units cancel out and we get xhr/30 and xhr/45
find the lcd, which is 90, and convert the 2 equations to obtain a denom of 90 in both</p>
<p>so 3Xhr/90 and 2Xhr/90
5Xhr/90 = Xhr/18<br>
since we know that the entire trip took 1 hr, we set Xhr/18 = 1hr
get rid of the hr units
X/18=1
so X = 18</p>
<p>check: if the trip is 18 miles long and she drives @ 30 mph, then she’ll take 18/30 hrs to get there = 3/5
driving at 45mph takes 18/45 hrs, or 2/5
3/5 + 2/5 = 1
so it makes sense</p>
<p>EDIT: yea math117’s method is easier, what a surprise</p>
<p>
</p>
<p>Reading the problem:
S1 = 45 and S2 =30 - Question is ONE WAY distance:</p>
<p>Answering:
(45*30) / (45+30) or 18.</p>
<p>Although some keep on insisting that it is best to set up systems of equations, this is exactly what TCB expects unprepared students to do as it IS a waste of time and a potential source of errors. </p>
<p>This problem is easily solved in fewer than 15 seconds.</p>
<p>yea I need to prepare for the math
xiggi, what math principle states that the answer can be found via (S1S2)/(S1+S2)?
I don’t have problems finishing on time but saving a minute or so is always useful</p>
<p>Thanks for all the answers! They seem so simple now. </p>
<p>@xiggi - So for any question like #18, multiply the speeds, and then divide by the addition of the speeds?</p>
<p>What you have here is an application of the harmomic mean formula. </p>
<p>Here’s an old post of mine on this that shows of the shortcut is still derived from the standard distance formula. </p>
<p>When looking at this type of problems, it is important to pay attention to details such as round trip versus one-way travel. In this case, the question was about 1/2 of the total trip. One cannot blindly plug the numbers without checking the facts stated in the problem.</p>
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