How do you sharpen a machete?
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I think a hatchet's more useful in the desert southwest. Check with the land agency where you're going as fire restrictions are common in the west. I usually skip fires entirely anymore. So often anymore you have to haul wood in with you anyway as there isn't any for gathering or gathering is prohibited.
As to food cooling for 6 days.
Ice Chests/Coolers are rated at a 2:1 ice to food ratio. So when you buy a cooler that says it will keep food cool for 5 days in 90 degree heat, they started with 2x as much ice as food. These are your average coolers. Remember that most campers do not pack them that way, but load them with food then squeeze in what ice they can. Then you get into expedition grade coolers that are tested the same way but rated for 10 days. They weigh 3X as much empty and have 2-3 inches of insulation. And cost about 4x as much. Costco is selling a 70 quart expedition grade cooler at $249 right now from Pelican. It's a pretty good buy.
Further, commercially frozen bags of ice aren't really all that cold. The Colorado River runners usually freeze their own. These are frozen much colder and harder so they last through the week of time on the river or more. Generally, block ice lasts longer than cube. But again, most commercial block ice is a bit of a cheat. They fill the block form with their cubes, then add water to fill in the spaces so it will freeze fast. But it never gets that hard or cold compared to what you can do on your own if you plan ahead.
How you pack your cooler matters if ice longevity is your goal. Plan your meals ahead. Pre-freeze whatever you can, Pack the food in the reverse order of how you'll use it. NEVER open the cooler until you know what you're going to remove and where it is. A paper guide to what you packed and how it's organized will help. Cross off things as you use them up. Keeping the cooler out of the sun helps. Keeping it in the AC part of your vehicle helps a lot while you drive. Keep it covered for added insulation and protection from sun and heat. Be aware that when you park your car, your cooler has to fight off the heat that builds in the sealed environment of the car.
Expedition grade coolers will do even better if you pre-chill them. Basically, they weigh enough that they have enough thermal mass to benefit from an overnight of a bag of commercial ice before loading and using. You have a little waste up front from the melt but gain more cooling power for the trip.
There are advocates who say not to drain the water from your cooler because that water is also cold. There are times I'm OK with this idea and times I'm not. Not everything does well getting wet. It will suck the heat out of your frozen food that's frozen cooler than the meltwater is. It gets into your packages and things. So balance your draining with what you have in the cooler. I drain more often than not. But do consider the drain water for clean up purposes if the raw meat juice hasn't got into it. On one river trip I took they mixed some drain water from a carefully packed cooler so they know it was still clean, to make a cold drink on day 4 of the trip. Cold was greatly apprecieated in the beverage by that point.
Know the limits of your cooler and its design so you can keep food safe. For the most part, this is about stratification of temperatures in your cooler. The average commercial ice chest will be right around freezing at the bottom and about 70 degrees at the top. So keep your vegetables high, your dairy and meat down low. I've not seen reports on how expedition grade coolers do at stratification, but it should be better.
Common ice chests are not animal proof. Racoons, skunks and more can and will violate your cooler if you leave it out. In bear country, you need to keep it locked up in your car and most camp sites require this practice as well. Most expedition grade coolers are bear-proof--at least to the bear opening it. But the bear might haul it off quite a ways before giving up on it.
There are some good meals that can be made with minimal or no refrigeration for later in you camping excursion. Red Flannel Hash from canned corned beef, canned potatoes, canned beets, fresh onions works pretty well. You can also just start with canned corned beef hash if you prefer. Many simple pasta dishes can be sauced with canned tomatoes cooked with things that don't need refrigeration like garlic and such. Similarly, the canned whole chickens, while easy to deride for most of us here provide a good quantity of gelatinous stock as well as meat to work with and needs no refrigeration. Canned bacon used to be pretty common in the 70s. Hard to find now. Works like a charm for camping situations. it's saltier than normal bacon so plan accordingly.
If you're going to be away from possible restocking on ice, that also puts you in a position of having to manage garbage for 6 days. That can be a trick in its own right.
I'd approach it differently, but what I want out of the experience might not be what you want.
Hauling camping and cooking supplies to a different spot every day includes a lot of setup and tear down time as well as hauling the heaviest of the gear everywhere. Plus, packable cooking gear is pretty weak.
I'd set up a base camp where I would do the major cooking and sleep every night. Plan for a fast breakfast. Because of the heat in AZ, leave int the AM twilight and eat a no-cook breakfast of some fruit, bagel, pre-made breafast burrito on the drive to the days hike or on the day's hike itself. Get away from camp for most of the day. Much lighter pack and you can go farther, see more, do more. Take a packable no-cook lunch. Return to camp to cook dinner and sleep.
http://www.wikihow.com/Clean-Dishes-on-a-Camping-Trip pay attention to steps 9-12 of the soap method. You don't discard solids with the grey water. You put those in your garbage. Keeps the bugs and animals from coming around so much.
I was a Scoutmaster for a few years. I saw Scouts do this. So the idea is in my head whether realistic for adults or not.