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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 \(y\)-intercept of a function \(t(x)\) is defined as the value of the function when the input \(x\) is \(0\).
  • For the given function \(t(x) = \frac{2,000}{900 + x}\), the value at \(x=0\) is \(t(0) = \frac{2,000}{900}\).
  • Since \(x=0\) indicates no wind, \(t(0)\) calculates the travel time of the plane under standard no-wind conditions.
  • The correct interpretation is: \(\boxed{\text{the time it takes the plane to travel without the tailwind}}\).

Explanation

  1. Analyzing the Function and the y-intercept

The problem asks us to determine the meaning of the \(y\)-intercept for the time function \(t(x) = \frac{2,000}{900 + x}\), where \(x\) represents the speed of a tailwind in kph. The \(y\)-intercept of any function \(f(x)\) occurs at the point where the input variable is zero, i.e., at \(x = 0\). Evaluating this function at \(x = 0\) will tell us the value of the function when there is no tailwind speed added to the plane's base speed.

  1. Evaluating and Interpreting the y-intercept

To find the \(y\)-intercept of \(t(x)\), we evaluate the function at \(x = 0\):
\[t(0) = \frac{2,000}{900 + 0} = \frac{2,000}{900} = \frac{20}{9} \approx 2.22 \text{ hours.}\]
This value represents the time taken to travel the fixed distance of \(2,000\) km when the added speed from the tailwind is exactly \(0\) kph. Since a tailwind speed of \(0\) kph means there is no wind at all, this value specifically represents the travel time for the plane under its standard speed of \(900\) kph.

  1. Identifying the Correct Option

Comparing our result with the provided options:

  • The time it takes the plane to travel without the tailwind: This matches our calculated meaning for \(t(0)\).
  • The speed of the plane when there is no tailwind present: This is \(900\) kph, which is the denominator, not the value of the time function itself.
  • The minimum amount of time it takes the plane to travel \(2,000\) km: This would occur as the tailwind speed \(x\) approaches infinity, not at \(x = 0\).
  • The time it takes the plane to travel when the speed is decreased by \(900\) kph: This would result in division by zero, as the speed becomes \(900 - 900 = 0\).

Therefore, the \(y\)-intercept represents the time it takes the plane to travel the given distance without any tailwind.

Examples

This concept of finding the intercept is fundamental in analyzing efficiency. For example, if you are planning a delivery route, the y-intercept of a cost-per-mile function can help you understand the fixed costs associated with the vehicle before you even start driving. Understanding these mathematical points allows businesses and engineers to model real-world performance accurately, from fuel consumption in flight to logistics planning on the ground.

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