<p>Can someone help me out with these 2 problems? They’re on page 91 of the Princeton Review 2006-2007 book. </p>
<li>A 384 square meter plot of land is to be enclosed by a fence and divided into two equal parts by another fence parallel to one pair of sides. What dimensions of the outer rectangle will minimize the amount of fence used?</li>
</ol>
<p>Answer: 16 x 24</p>
<li>A swimmer is at a point 500 m from the closest point on a straight shoreline. She needs to reach a cottage located 1800 m down shore from the closest point. If she swims at 4 m/s and she walks 6 m/s how far from the cottage should she come ashore so as to arrive at the cottage in the shortest time?</li>
</ol>
<p>Answer: x=1350 m</p>
<p>Oh and just this one problem to on page 12 </p>
<li>Find lim (sin(x + h) -sinx)/h
h>0<br></li>
</ol>
<p>Answer: cosx</p>
<p>I just can’t figure these out. ANY help would be GREATLY appreciated thanks!</p>
<p>I used a calculator to do the arithmetic on this but here goes:</p>
<p>3) This is an optimization question; you are trying to optimize Fencing. Drawing the picture out shows you that the fencing you'd need is 3(width) + 2(length). You're also given that the area is 384. </p>
<p>lw=384
F=2l+3w</p>
<p>Substituting 384/w for length, you'd get
F=3w+ 768/w</p>
<p>Derive that to get F'= 3- 768/w^2</p>
<p>Set that equal to 0 to find the critical points. (-16 and 16)
-16 isn't possible, so W=16</p>
<p>Plug in 16 to 384=lw, and L=24</p>
<p>16x24</p>
<p>I'm working on #5 now, but #29 is the limit definition of the derivative.</p>
<p>limit f(x-h)-f(x)
h->0 h</p>
<p>so your sin(x) is f(x). Derive sin(x) and you'll get cos(x)</p>
<p>hey thanks a lot for the help, I REALLY appreciate. can anybody else help me out with #5?</p>
<p>It might help to draw this, since I can't explain worth anything. I think that this is an optimization problem. You have a vertical line, whose total length is 1800. Perpendicular to this, you have another line, length 500. At the end of this perpendicular line farthest from the vertical line is the swimmer. If you draw a line from the swimmer to the shore, towards the cottage but not all the way, you'll get a triangle. Now make the distance from the cottage to the intersection of this new line (the swimmer's path) with the vertical line 1800 - x. Make the remaining section of the vertical line (one of the legs of the right triangle) x. Now, if you calculate the distance that the swimmer will have to travel (by the Pythagorean Theorem) you get sqrt(x^2 + 500^2) + 1800 - 8. Since time is distance divided by speed, you get t(x) = sqrt(x^2 + 500^2)/4 + (1800 - x)/6. Take the derivative and set it to zero. You should get x = 200*sqrt(5). Subtract this distance from 1800 to get about 1352.8.</p>