We now have obtained or defined the variables necessary to formulate the energy required to hike a trail in the Black Hills area. As we explain above, we know these calculations will leave us with two significant figures at best. We have explained also that if we can arrive at a precision within 5% to 10%, we will find these calculations useful in planning a hike. So we will shape our formula with a degree of skepticism and allow the future to correct our work.

There are two more parameters we have used to confine our defined values. We have often hiked the Willow Creek to Harney Peak trail and timed our hike. We have completed the hike in 5.5 hours, but more often it took 6.5 to 7.0 hours. If our model and hypothesis resulted in a nine-hour hike for that trail, we remodeled it within the guidelines we have noted. Additionally, we have kept calorie-intake records and a log of all hikes within a year’s time that suggest our calorie consumption for the Willow Creek to Harney hike is within the range of 2600 to 3400 calories. We would be suspicious if this formula found a result outside that range, which it did not. Our previous formula did, and the defined values we unintentionally entered were redefined.

What we are not changing is the new formula itself, the simpler formula we used previously, the ACSM statistics (except as noted above), the elevation changes provided by our review of topographic maps, nor the mileage we have previously posted. Most notably, the defined values we found it necessary to adjust for our new formula and for the old were the sustained rate of climbing, and the calorie consumption per mile of level walk. This yielded a new value for the simpler formula of about 280-feet of climbing being equivalent in energy consumption to one mile of walking.

When we returned to the simpler formula and replaced the two defined variables, with new values—one derived from ACSM tables (as described above) and the other derived from calculations of the new formula, we found that the new and the older, simple formula produced calorie consumption figures for every trail within 99% precision between formulas.

This surprising result does not in any way prophetically confirm the validity of the final calorie figure. What it does suggest is that the simple formula we began with may be arrived at by mathematically reducing the variables of our new formula. That remains to be seen when our curiosity again persuades us to resume this project.

If the new and old formulas can be so closely synchronized, what have we gained with the new formula? The new formula based upon ACSM tables forced us to more closely confine our defined values within an experimental range. The new value for calories consumed while hiking a level trail has substantially altered the hierarchy of trails with regard to energy consumption. Longer trails now require more energy than was supposed and higher-climbing trails require less. We now must climb 280 feet instead of 167 feet to expend as much energy in climbing as when walking one mile on a level trail. Our old formula is as precise as the new formula, but only after the new formula has caused us to redefine basic values in the old.

To the extent that our hypothetical model is correct, a simple rule of thumb would be that it takes about as much energy (but no more) to walk a mile and a half around a 280-foot high hill as it does to walk a half-mile over the hill. But of course the trail goes around the hill to reduce erosion. In our new tabulation, for instance, segments of the Mickelson Trail, though seemingly rather flat, rank higher in energy consumption relative to Harney Peak trails than we had supposed—even if hiking some of the Harney peak trails still consumes more energy.

So we will conclude this discussion by delineating the new and older formulas and by tabulating the defined values as given above. Both formulas will here employ the notation used in Microsoft Excel spreadsheets. Column A is blank. Column B is reserved for the name of the trail. An asterick (*) signifies multiplication. We began the formula on line 9 of the spreadsheet and define the columns of variables with the letters that follow:

C=ENTER: Accumulated climb in feet going away from the trailhead to the end of the trail.

D=ENTER: Accumulated climb in feet from the end of the trail back to the trailhead.

E=ENTER: Length of a trail in miles one-way (or half the length of a loop trail).

F=CALCULATE: Average slope out (not used in formula):
(C9/(E9*5280))

G= CALCULATE: Average slope back (not used in formula):
(D9/(E9*5280))

H= CALCULATE: Miles traveled to climb C at 14% grade (a rise of 1.5 inches for each 10.5-inch step):
((C9*7)/5280)

I= CALCULATE: Miles traveled to climb D at 14% grade (a rise of 1.5 inches for each 10.5-inch step):
((D9*7)/5280)

J= CALCULATE: Trail miles remaining for level and downhill walking combined:
((E9*2)-(H9+I9))

K= CALCULATE: Miles hiked downhill (same as miles hiked uphill):
(H9+I9)

L= CALCULATE: Miles hiked on level trail:
(J9-K9)

M= CALCULATE: Hours spent walking uphill at 700 feet per hour:
((C9+D9)/700)

N= CALCULATE: Hours spent walking downhill at 2.5 miles per hour:
(K9/2.5)

O= CALCULATE: Hours spent walking on level trail at 2.5 miles per hour:
(L9/2.5)

P= CALCULATE: Total Calorie Consumption for Entire Hike for 130-pound person carrying a 15-pound pack:
((M9*464)+(N9*322)+(O9*345))

The entire NEW formula for P calculated from only columns C, D, and E, can be entered into a single column as:

(((C9+D9)/700)*464)+(((((C9*7)/5280)+((D9*7)/5280))/2.5)*322)+(((((E9*2)-(((C9*7)/5280)+((D9*7)/5280)))-(((C9*7)/5280)+((D9*7)/5280)))/2.5)*345)

The OLDER simple formula for P (revised) calculated from only columns C, D, and E, can be entered into a single column as:
(((C9/280)+(D9/280)+(2*E9))*135)

Written out by hand rather than for a spreadsheet, and with units of measurement indicated, the new formula resolves as follows:
    

Both new and older formulas involve only the four basic functions of mathmatics: addition, subtraction, multiplication, and division. No higher mathematics are involved. It so happens that it has taken more time to show and explain our work than to do it in the first place, but the time has added to clarity of thought, exposed a trigonometric factor we overlooked, and promoted our ability to revise and correct. That time we suspect will be returned to us as the weeks pass and we prepare our hikes.

Return to Calorie Consumption Table