The First Ski Club Van Has Been On The Road For 20 Minutes
The First Ski Club Van Has Been On The Road For 20 Minutes And The Se
The problem involves understanding the time each van has spent on the road, expressing that mathematically, forming ratios, and calculating those ratios after specific periods of additional time. First, we are given initial conditions: the first van has been driving for 20 minutes, and the second for 35 minutes. Then, we will incorporate an additional t minutes, find the ratio of their times, and evaluate that ratio after 60 and 200 minutes.
Part A: Expressions for the total time each van has traveled after an additional t minutes
The first van has been on the road for 20 minutes initially. After an additional t minutes, the total time the first van has spent on the road is:
\[ T_1(t) = 20 + t \]
Similarly, the second van has been on the road for 35 minutes initially. After t additional minutes, the total for the second van is:
\[ T_2(t) = 35 + t \]
These expressions precisely model the total time each van has been on the road after t minutes.
Part B: Ratio of the first van's time to the second van's time and evaluating after specific times
The ratio of the first van's total time to that of the second van's at any given t is:
\[ R(t) = \frac{T_1(t)}{T_2(t)} = \frac{20 + t}{35 + t} \]
This ratio provides a comparison of their relative times on the road at any moment considering an extra t minutes.
Using long division, we can rewrite this as an expression:
Dividing numerator by denominator:
\[ \frac{20 + t}{35 + t} \]
In long division form:
- When t = 0:
\[ \frac{20}{35} = \frac{4}{7} \approx 0.5714 \]
- For a general t, note that as t becomes large, the ratio approaches 1, because the added t dominates the initial constants.
Expressed as a mixed form:
\[ R(t) = 1 - \frac{15}{35 + t} \]
This is derived from rewriting:
\[ \frac{20 + t}{35 + t} = \frac{(35 + t) - 15}{35 + t} = 1 - \frac{15}{35 + t} \]
Calculating the ratio after 60 minutes and 200 minutes
After 60 minutes:
Set \( t = 60 - 20 = 40 \) minutes for the first van (since it initially traveled 20 minutes). But the problem asks for total times after 60 minutes overall.
- First van:
\[ T_1 = 20 + t = 20 + 40 = 60 \ \text{minutes} \]
- Second van:
\[ T_2 = 35 + t = 35 + 25 = 60 \ \text{minutes} \]
here, t is 40 minutes for the first van to reach 60 minutes total; for the second, t is 25 minutes (since it initially traveled 35, adding 25 gives 60). However, the ratio expression applies at the same t for both after the same additional time t, not necessarily at a total of 60 minutes for each individually.
Alternatively, since the initial times are fixed, and the problem wants the ratio after a total of 60 and 200 minutes, the best approach is to find the time t such that:
\[ T_1(t) = 20 + t = 60 \implies t = 40 \]
Similarly,
\[ T_2(t) = 35 + t \]
to get their total times at 60 minutes:
\[ T_2(t) = 35 + 40 = 75 \ \text{minutes} \]
which exceeds 60. So, to compare ratios after 60 minutes total, at t:
- First van: 20 + t = 60 → t = 40
- Second van: 35 + t = ?
Since t = 40, second van has:
\[ T_2 = 35 + 40 = 75 \]
Thus, the ratio at t=40:
\[ R(40) = \frac{20 + 40}{35 + 40} = \frac{60}{75} = \frac{4}{5} = 0.8 \]
Similarly, after 200 minutes:
- First van:
\[ T_1 = 20 + t \]
- Second van:
\[ T_2 = 35 + t \]
Set \( T_1 = 200 \):
\[ 20 + t = 200 \implies t = 180 \]
Calculate \( T_2 \) at \( t = 180 \):
\[ T_2 = 35 + 180 = 215 \]
Calculate ratio:
\[ R(180) = \frac{20 + 180}{35 + 180} = \frac{200}{215} \approx 0.9302 \]
Conclusion
The ratio advances from approximately 0.5714 at the start to nearly 1 as t increases, exemplified by ratios of 0.8 and approximately 0.93 after 60 and 200 minutes respectively. This signifies that over time, both vans' relative durations become more comparable, aligning towards equality as the initial disparity diminishes in significance compared to the added journey time.
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Paper For Above instruction
Introduction
Travel time analysis between two vehicles offers insights into relative progress, scheduling, and efficiency. In this context, we examine two ski club vans that have been traveling for different initial durations, and explore how their relative times evolve as they continue on their journeys. By developing algebraic expressions and evaluating ratios after specific periods, we gain a clearer understanding of their comparative travel statuses.
Modeling Initial Conditions and Additional Travel Time
The initial conditions specify that the first van has traveled for 20 minutes, while the second has traveled for 35 minutes. To model their total travel times after an additional t minutes, we assign variables:
\[
T_1(t) = 20 + t \\
T_2(t) = 35 + t
\]
These equations assume continuous, uniform travel without stops or delays, simplifying real-world complexity but providing a useful theoretical framework. The addition of t minutes signifies ongoing travel, allowing us to analyze their relative progress dynamically.
Formulating the Ratio of Travel Times
The core comparison involves the ratio of the first van’s travel time to that of the second:
\[
R(t) = \frac{T_1(t)}{T_2(t)} = \frac{20 + t}{35 + t}
\]
This ratio indicates how proportionally the vans are progressing. Notably, as t increases, the ratio approaches 1, illustrating that the disparity in initial travel times becomes less significant compared to ongoing travel.
Expressing this ratio in a different form provides additional insights. Applying long division:
\[
\frac{20 + t}{35 + t}
\]
Rearranged, it becomes:
\[
R(t) = 1 - \frac{15}{35 + t}
\]
This form emphasizes the diminishing difference as t increases, illustrating convergence toward a ratio of 1 over time.
Calculating Ratios After Specific Total Travel Durations
To determine the relative travel times at specific points—say, when one or both vans have traveled for 60 or 200 minutes—we solve for t in the expressions:
- For total of 60 minutes for the first van:
\[
20 + t = 60 \implies t = 40
\]
Calculating the second van’s time:
\[
T_2 = 35 + 40 = 75
\]
The ratio at this point:
\[
R(40) = \frac{60}{75} = \frac{4}{5} = 0.8
\]
- For total of 60 minutes for the second van:
\[
35 + t = 60 \implies t = 25
\]
Calculating the first van’s time at this t:
\[
T_1 = 20 + 25 = 45
\]
which yields a ratio:
\[
R(25) = \frac{45}{60} = \frac{3}{4} = 0.75
\]
Similarly, for a total of 200 minutes:
\[
20 + t = 200 \implies t = 180
\]
The second van’s total time at t = 180:
\[
T_2 = 35 + 180 = 215
\]
Corresponding ratio:
\[
R(180) = \frac{200}{215} \approx 0.9302
\]
These calculations illustrate that as both vans continue traveling, their travel times increasingly resemble each other proportionally.
Implications and Conclusions
Analyzing the ratio of travel times over the span of their journeys reveals a trend toward synchronization. Initially, the disparity reflects their different starting points but diminishes as they continue traveling. The approach toward a ratio of 1 suggests that over long periods, initial differences become negligible in proportion, aligning with broader concepts of convergence and relative progress in motion analysis.
This methodological framework is applicable in diverse scenarios such as vehicle fleet management, race analysis, and scheduling, wherever understanding relative progress over time is critical.
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