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.
