Let S Spose We Re in Charge of an Area of Forest Which Is 50 Km by 50 Km Square

MATH 136 CALCULUS 1 From a management perspective, we consider the total area used up by SPRING 2010 such a plan to be the area we cleared for the firebreaks plus the area within one stand (which could potentially burn down), i.e.

Total Area = area of firebreaks + area of one stand ONLY YOU CALCULUS CAN PREVENT FOREST FIRES **Find the total area used by four firebreaks. We want to find the perfect number of firebreaks to cut – that is, the number of firebreaks which will minimize the total area lost by Get into a group of no bigger than 5. Put your heads together and make firebreaks and potential fires. Toward that end, it happen! Only turn in one solution per group. You’re answering the questions that have **’s in front of them. The ones with ****’s in front of **Let x be the number of firebreaks and find a general function for the them are extra important. total area lost in terms of x, call it A(x). One management technique for reducing the damage done by forest We’re professional Calculusers, so we know that the minimum of a fires is to have a person dress up in a bear costume. Another technique function (we call them “valleys” sometimes) can only occur at the is to use what are called firebreaks – these are basically strips of land critical points of a function (i.e. the x-values at which the derivative is that are cleared through a forest with the hope that, if a fire starts on one zero or undefined). So, if we want to find the minimum of our area side of the strip, it won’t be able to jump across the firebreak and function, we need to continue burning on the other side. If we cut lots of firebreaks, fires will be contained to small areas – but that means clearing large swaths of **Find the critical point(s) of the general area function you just defined. trees. So we’re looking for the optimal balance. ****How many firebreaks will minimize the area lost under this Let’s spose we’re in charge of an area of forest which is 50 km by 50 km management strategy? square. Clearing firebreaks turns out to be a lot of fun, so now we’re going to cut First, we’re going to try to cut firebreaks which are simply strips of land them in both directions, like this: which stretch across the width of our forest at even intervals. We also put one firebreak at one (and only one) boundary of our forest. Each of our firebreaks is 0.01 km wide (always). If we cut four such firebreaks, it’ll look something like this:

Note that there are the same number of firebreaks in each direction and so the resulting stands are squares. Also, there are firebreaks on two (and only two) boundaries. Again, the idea is that if a fire starts in one “stand,” those trees will burn down, but the fire won’t jump across our firebreak, so the rest of the trees will be safe. **Let x be the number of firebreaks and find a general function for the total area lost in terms of x, call it A(x). **Find the critical point(s) of the general area function you just defined. ****How many firebreaks will minimize the area lost under this management strategy?