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A plane travels 2,000 kilometers at a speed of 900 kilometers per hour (kph) with no wind. When a tailwind is present, the plane's speed increases by $x$ kilometers per hour. The time it takes the plane to travel the same distance with the tailwind, $t(x)$, is defined by the function $t(x)=\frac{2,000}{900+x}$. What is the meaning of the $y$-intercept for this function?


AThe time it takes the plane to travel without the tailwind
BThe speed of the plane when there is no tailwind present
CThe minimum amount of time it takes the plane to travel 2,000 km
DThe time it takes the plane to travel when the speed is decreased by 900 kph

Verified Answer :

  • The function \(t(x) = \frac{2000}{900 + x}\) models travel time with a tailwind of speed \(x\).
  • The \(y\)-intercept occurs when the input \(x = 0\), corresponding to a condition with no tailwind.
  • Evaluating the function at zero, \(t(0) = \frac{2000}{900} = \frac{20}{9}\) hours, which represents the travel time without any wind.
  • The correct meaning is \(\boxed{\text{the time it takes the plane to travel without the tailwind}}\).

Explanation

  1. Problem Analysis and Definitions

We are given that a plane travels a distance of \(2,000\) kilometers. In the absence of wind, its speed is \(900\) kilometers per hour (kph). When a tailwind of speed \(x\) kph is present, the effective speed of the plane becomes \(900 + x\) kph. The time \(t(x)\) required to travel the \(2,000\) km is given by the function:
\[t(x) = \frac{2,000}{900 + x}\]
The \(y\)-intercept of a function \(f(x)\) represents the value of the function when \(x = 0\). In this physical context, \(x\) represents the speed of the tailwind. Setting \(x = 0\) signifies that there is no tailwind present.

  1. Evaluating the y-intercept

To find the \(y\)-intercept, we evaluate the function \(t(x)\) at \(x = 0\):
\[t(0) = \frac{2,000}{900 + 0} = \frac{2,000}{900} = \frac{20}{9} \text{ hours.}\]
This value represents the time taken for the plane to travel the \(2,000\) km distance when the tailwind speed is \(0\) kph, which is precisely the situation where there is no wind at all.

  1. Conclusion

We now compare our finding with the given options:

  1. "the time it takes the plane to travel without the tailwind": This aligns with our calculation where \(x=0\) implies no tailwind.
  2. "the speed of the plane when there is no tailwind present": This is incorrect because \(t(x)\) measures time, not speed.
  3. "the minimum amount of time it takes the plane to travel 2,000 km": This is incorrect because the time decreases as the tailwind speed \(x\) increases.
  4. "the time it takes the plane to travel when the speed is decreased by 900 kph": This is incorrect as it describes a different physical scenario.

Therefore, the \(y\)-intercept corresponds to the time it takes the plane to travel without the tailwind. The final answer is \(\boxed{\text{the time it takes the plane to travel without the tailwind}}\).

Examples

Understanding the y-intercept in functional relationships is crucial in physics and aviation. For instance, if you are calculating fuel consumption based on vehicle speed, the y-intercept often represents the fuel consumption while idling (at zero speed). By interpreting these intercepts, engineers and analysts can determine baseline performance metrics, helping to optimize systems for efficiency and safety in real-world travel scenarios.

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