To make the Chinese dish "dragon's beard noodles," fold a 5 ft. strand of dough in half. Stretch the dough back to its original length so that two thinner strands are formed. Repeat this folding and stretching process over and over to produce increasing numbers of progressively thinner noodles.
a) Write an equation that gives the combined length L in feet of all the noodles after folding and stretching the dough x times. How did you get your answer?
This problem totally confuses me out of my pants.
The total length of strands after one split-an-stretch is 2* 5 ft = 20 ft
After doing it twice, it is 40 ft
After n spits and stretches, it is
10*2^n long
isn't 2*5ft eqaul 10 feet, not 20 ?
After doing it twice would be 20 ?
I agree that 2 * 5 ft = 10.
I would go with 5 * 2^n, since you started out with 5 ft.
I hope this helps a little more. Thanks for asking.
To derive an equation for the combined length L of all the noodles after folding and stretching the dough x times, let's break down the process step by step:
1. Start with a 5 ft. strand of dough.
2. Fold the dough in half, resulting in two strands of equal length, each measuring 5 ft. / 2 = 2.5 ft.
3. Stretch the two strands back to their original length of 5 ft.
4. Repeat the folding and stretching process x times.
Let's consider the length of the noodles after each fold and stretch:
- After the first fold and stretch: We have two strands, each measuring 5 ft. / 2 = 2.5 ft. The total length is 2 * 2.5 ft. = 5 ft.
- After the second fold and stretch: Each of the two strands is folded and stretched again. So we now have four strands, each measuring 2.5 ft. / 2 = 1.25 ft. The total length is 4 * 1.25 ft. = 5 ft.
- After the third fold and stretch: Each of the four strands is folded and stretched again. So we now have eight strands, each measuring 1.25 ft. / 2 = 0.625 ft. The total length is 8 * 0.625 ft. = 5 ft.
- And so on...
We can observe a pattern emerging:
- After the first fold and stretch: Total length = 5 ft.
- After the second fold and stretch: Total length = 5 ft.
- After the third fold and stretch: Total length = 5 ft.
- ...
No matter how many times the dough is folded and stretched, the total length remains constant at 5 ft.
Therefore, the equation for the combined length L in feet of all the noodles after folding and stretching the dough x times is:
L = 5 ft.
The explanation shows that no matter how many times the dough is folded and stretched, the combined length of the noodles will always be 5 ft.