To determine the distance between the refreshment stands, we first note that there are stands at both the beginning and the end of the 34-mile route, plus 8 additional stands along the route.
This gives us a total of:
- 1 stand at the start
- 8 stands along the route
- 1 stand at the end
So, we have a total of \(1 + 8 + 1 = 10\) stands.
Next, we need to calculate the number of gaps between the stands. Since the first stand is at the start and the last stand is at the end, there are \(10 - 1 = 9\) gaps.
Now we will convert the route length from miles to feet. There are 5280 feet in a mile, so:
\[ 34 \text{ miles} \times 5280 \text{ feet/mile} = 179,520 \text{ feet} \]
Now we divide the total length of the route (in feet) by the number of gaps to find the distance between the stands:
\[ \text{Distance between stands} = \frac{179,520 \text{ feet}}{9 \text{ gaps}} = 19,947.78 \text{ feet} \]
So, rounding to the nearest whole number, the distance between the refreshment stands is approximately 19,948 feet.
Therefore, the refreshment stands are 19,948 feet apart.