<p>I know this isn't the right place to be putting this, but there are tons of math buffs on here, and I just can't seem to get these problems. I'd appreciate any help I can get. Here goes nothing:</p>
<p>1) Jane is 2 mi offshore in a boat and wishes to reach a costal village 6 mi down a straight shoreline from the point nearest the boat. She can row 2 mph and can walk 5 mph. Where should she land her boat to reach the village in the least amount of time</p>
<p>2)
You operate a tour service that offers the following rates:
-$200 per person if 50 people (the minimum number to book the tour) go on the tour
-For each additional person, up to a max of 80 people total, the rate per person is reduced by $2
It costs $6000 (a fixed cost) plus $32 per person to conduct the tour. How many people does it take to maximize your profit.</p>
<p>Take the equation for earnings(I) versus passengers(X):
I = X(200 - (2X - 100))
= X(300 - 2x)
= 300x - 2x^2</p>
<p>The equation for ammount spent(S):<br>
S = 6000 + 32X</p>
<p>The profit(P) will be:<br>
P = I - S
P = 300x - 2x^2 - 6000 - 32x
P = -2x^2 + 268x - 6000</p>
<p>Here is where the calculus comes into play(or you can let the calculator do it for you... which is my favorite method); since it is a parabola opening down(-2x^2), you know the the maximum will be at the point when the derivative is equal to 0. So, lets find the derivative of P.</p>
<p>P = -2x^2 + 268x - 6000
P' = -4x + 268 (that was easy)</p>
<p>Now set it equal to zero</p>
<p>0 = -4x+268
4x = 268
x = 67</p>
<p>Since X falls between 50 and 80, we know we're alright there.</p>
<p>If you have any questions about where I got the equations or why I did anything just ask I'll check back in a bit, but I think I did a fairly good job explaining it.</p>