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To estimate the distance traveled by the vehicle during the 55-second period, we will use two different methods: one using the velocities at the beginning of the time intervals and the other using the velocities at the end of the time intervals.
1. Using Velocities at the Beginning of the Time Intervals:
- We will calculate the distance traveled during each interval by multiplying the velocity at the beginning of the interval by the duration of the interval.
- Then, we will sum up these distances to get the total distance traveled.
Interval 0-11 seconds:
[tex]\[ \text{Distance} = 37 \, \text{ft/s} \times (11 - 0)\,\text{s} = 37 \, \text{ft/s} \times 11 \, \text{s} = 407 \, \text{ft} \][/tex]
Interval 11-22 seconds:
[tex]\[ \text{Distance} = 36 \, \text{ft/s} \times (22 - 11)\,\text{s} = 36 \, \text{ft/s} \times 11 \, \text{s} = 396 \, \text{ft} \][/tex]
Interval 22-33 seconds:
[tex]\[ \text{Distance} = 31 \, \text{ft/s} \times (33 - 22)\,\text{s} = 31 \, \text{ft/s} \times 11 \, \text{s} = 341 \, \text{ft} \][/tex]
Interval 33-44 seconds:
[tex]\[ \text{Distance} = 33 \, \text{ft/s} \times (44 - 33)\,\text{s} = 33 \, \text{ft/s} \times 11 \, \text{s} = 363 \, \text{ft} \][/tex]
Interval 44-55 seconds:
[tex]\[ \text{Distance} = 27 \, \text{ft/s} \times (55 - 44)\,\text{s} = 27 \, \text{ft/s} \times 11 \, \text{s} = 297 \, \text{ft} \][/tex]
Summing these distances:
[tex]\[ \text{Total distance} \approx 407 + 396 + 341 + 363 + 297 = 1804 \, \text{ft} \][/tex]
2. Using Velocities at the End of the Time Intervals:
- We will calculate the distance traveled during each interval by multiplying the velocity at the end of the interval by the duration of the interval.
- Then, we will sum up these distances to get the total distance traveled.
Interval 0-11 seconds:
[tex]\[ \text{Distance} = 36 \, \text{ft/s} \times (11 - 0)\,\text{s} = 36 \, \text{ft/s} \times 11 \, \text{s} = 396 \, \text{ft} \][/tex]
Interval 11-22 seconds:
[tex]\[ \text{Distance} = 31 \, \text{ft/s} \times (22 - 11)\,\text{s} = 31 \, \text{ft/s} \times 11 \, \text{s} = 341 \, \text{ft} \][/tex]
Interval 22-33 seconds:
[tex]\[ \text{Distance} = 33 \, \text{ft/s} \times (33 - 22)\,\text{s} = 33 \, \text{ft/s} \times 11 \, \text{s} = 363 \, \text{ft} \][/tex]
Interval 33-44 seconds:
[tex]\[ \text{Distance} = 27 \, \text{ft/s} \times (44 - 33)\,\text{s} = 27 \, \text{ft/s} \times 11 \, \text{s} = 297 \, \text{ft} \][/tex]
Interval 44-55 seconds:
[tex]\[ \text{Distance} = 47 \, \text{ft/s} \times (55 - 44)\,\text{s} = 47 \, \text{ft/s} \times 11 \, \text{s} = 517 \, \text{ft} \][/tex]
Summing these distances:
[tex]\[ \text{Total distance} \approx 396 + 341 + 363 + 297 + 517 = 1914 \, \text{ft} \][/tex]
Therefore, the estimated distances traveled by the vehicle over the 55-second period are:
- Using velocities at the beginning of the time intervals: [tex]\(\boxed{1804 \, \text{feet}}\)[/tex]
- Using velocities at the end of the time intervals: [tex]\(\boxed{1914 \, \text{feet}}\)[/tex]
1. Using Velocities at the Beginning of the Time Intervals:
- We will calculate the distance traveled during each interval by multiplying the velocity at the beginning of the interval by the duration of the interval.
- Then, we will sum up these distances to get the total distance traveled.
Interval 0-11 seconds:
[tex]\[ \text{Distance} = 37 \, \text{ft/s} \times (11 - 0)\,\text{s} = 37 \, \text{ft/s} \times 11 \, \text{s} = 407 \, \text{ft} \][/tex]
Interval 11-22 seconds:
[tex]\[ \text{Distance} = 36 \, \text{ft/s} \times (22 - 11)\,\text{s} = 36 \, \text{ft/s} \times 11 \, \text{s} = 396 \, \text{ft} \][/tex]
Interval 22-33 seconds:
[tex]\[ \text{Distance} = 31 \, \text{ft/s} \times (33 - 22)\,\text{s} = 31 \, \text{ft/s} \times 11 \, \text{s} = 341 \, \text{ft} \][/tex]
Interval 33-44 seconds:
[tex]\[ \text{Distance} = 33 \, \text{ft/s} \times (44 - 33)\,\text{s} = 33 \, \text{ft/s} \times 11 \, \text{s} = 363 \, \text{ft} \][/tex]
Interval 44-55 seconds:
[tex]\[ \text{Distance} = 27 \, \text{ft/s} \times (55 - 44)\,\text{s} = 27 \, \text{ft/s} \times 11 \, \text{s} = 297 \, \text{ft} \][/tex]
Summing these distances:
[tex]\[ \text{Total distance} \approx 407 + 396 + 341 + 363 + 297 = 1804 \, \text{ft} \][/tex]
2. Using Velocities at the End of the Time Intervals:
- We will calculate the distance traveled during each interval by multiplying the velocity at the end of the interval by the duration of the interval.
- Then, we will sum up these distances to get the total distance traveled.
Interval 0-11 seconds:
[tex]\[ \text{Distance} = 36 \, \text{ft/s} \times (11 - 0)\,\text{s} = 36 \, \text{ft/s} \times 11 \, \text{s} = 396 \, \text{ft} \][/tex]
Interval 11-22 seconds:
[tex]\[ \text{Distance} = 31 \, \text{ft/s} \times (22 - 11)\,\text{s} = 31 \, \text{ft/s} \times 11 \, \text{s} = 341 \, \text{ft} \][/tex]
Interval 22-33 seconds:
[tex]\[ \text{Distance} = 33 \, \text{ft/s} \times (33 - 22)\,\text{s} = 33 \, \text{ft/s} \times 11 \, \text{s} = 363 \, \text{ft} \][/tex]
Interval 33-44 seconds:
[tex]\[ \text{Distance} = 27 \, \text{ft/s} \times (44 - 33)\,\text{s} = 27 \, \text{ft/s} \times 11 \, \text{s} = 297 \, \text{ft} \][/tex]
Interval 44-55 seconds:
[tex]\[ \text{Distance} = 47 \, \text{ft/s} \times (55 - 44)\,\text{s} = 47 \, \text{ft/s} \times 11 \, \text{s} = 517 \, \text{ft} \][/tex]
Summing these distances:
[tex]\[ \text{Total distance} \approx 396 + 341 + 363 + 297 + 517 = 1914 \, \text{ft} \][/tex]
Therefore, the estimated distances traveled by the vehicle over the 55-second period are:
- Using velocities at the beginning of the time intervals: [tex]\(\boxed{1804 \, \text{feet}}\)[/tex]
- Using velocities at the end of the time intervals: [tex]\(\boxed{1914 \, \text{feet}}\)[/tex]
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