To determine the total distance fallen by the bungee jumper after five falls, we need to analyze the given pattern in the distances of each fall, recognize it, and then use this pattern to find the distances for the fourth and fifth falls. Finally, we will sum all these distances.
1. Observe the pattern in the fall distances:
- First fall: 97.2 feet
- Second fall: 64.8 feet
- Third fall: 43.2 feet
2. Recognize that the sequence of distances is decreasing in a specific pattern. We can see that this is a geometric sequence where each distance is reduced by a common ratio from the previous distance.
To find the common ratio \( r \):
[tex]\[ r = \dfrac{\text{Distance of 2nd fall}}{\text{Distance of 1st fall}} = \dfrac{64.8}{97.2} \][/tex]
3. Calculate the common ratio \( r \):
[tex]\[ r = \dfrac{64.8}{97.2} = 0.6667 \][/tex]
4. Using this common ratio, find the fourth and fifth falls:
- Fourth fall:
[tex]\[ \text{Fourth fall} = 43.2 \times 0.6667 = 28.8 \, \text{feet} \][/tex]
- Fifth fall:
[tex]\[ \text{Fifth fall} = 28.8 \times 0.6667 = 19.2 \, \text{feet} \][/tex]
5. List all the fall distances:
- First fall: 97.2 feet
- Second fall: 64.8 feet
- Third fall: 43.2 feet
- Fourth fall: 28.8 feet
- Fifth fall: 19.2 feet
6. Sum the total distance fallen:
[tex]\[ \text{Total distance} = 97.2 + 64.8 + 43.2 + 28.8 + 19.2 \][/tex]
7. Calculate the total:
[tex]\[ \text{Total distance} = 97.2 + 64.8 + 43.2 + 28.8 + 19.2 = 253.2 \, \text{feet} \][/tex]
Therefore, the total distance the bungee jumper drops after five falls is \( 253.2 \, \text{feet} \).
Among the given options, the correct answer is:
[tex]\[ \boxed{253.2 \, \text{ft}} \][/tex]
