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 [tex]$x[tex]$[/tex] kilometers per hour. The time it takes the plane to travel the same distance with the tailwind, [tex]$[/tex]t(x)[tex]$[/tex], is defined by the function [tex]$[/tex]t(x)=\frac{2,000}{900+x}$[/tex].

What is the meaning of the [tex]$y$[/tex]-intercept for this function?

A. The time it takes the plane to travel without the tailwind.
B. The speed of the plane when there is no tailwind present.
C. The minimum amount of time it takes the plane to travel 2,000 km.
D. The time it takes the plane to travel when the speed is decreased by 900 kph.



Answer :

Let's take a look at the function \( t(x) = \frac{2,000}{900+x} \) step-by-step to understand what the \( y \)-intercept represents:

1. Understanding the Variables in the Function:
- \( 2,000 \) kilometers is the distance the plane travels.
- \( 900 \) kph is the speed of the plane without any wind.
- \( x \) represents the increase in speed due to a tailwind, in kilometers per hour (kph).

2. Function Simplification:
- The function \( t(x) = \frac{2,000}{900 + x} \) gives the time it takes for the plane to travel 2,000 kilometers when there is a tailwind increasing the speed by \( x \) kilometers per hour.

3. Finding the \( y \)-intercept:
- The \( y \)-intercept of a function is the value of the function when \( x = 0 \).
- Substituting \( x = 0 \) into the function \( t(x) \):

[tex]\[ t(0) = \frac{2,000}{900 + 0} = \frac{2,000}{900} \][/tex]

4. Calculation for \( y \)-intercept:
- Calculating the value of \( t(0) \):

[tex]\[ t(0) = \frac{2,000}{900} \approx 2.2222222222222223 \text{ hours} \][/tex]

5. Interpreting the \( y \)-intercept:
- The \( y \)-intercept, which is approximately 2.222 hours, represents the time it takes for the plane to travel the 2,000 kilometers when there is no tailwind.

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

Thus, the correct answer is:
the time it takes the plane to travel without the tailwind.