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Bot GPT 4-o mini To solve this problem, we need to determine how far away from the target the jet should drop the aid package.
### Step 1: Calculate the Time of Flight
The aid package will fall under the influence of gravity, and we can use the formula for the time of flight for an object falling from a certain height. The height \( h \) the package is dropped from is 1400 m, and the equation for the distance fallen under constant acceleration is:
\[
h = \frac{1}{2} g t^2
\]
Where:
- \( h = 1400 \, m \) (height),
- \( g = 9.81 \, m/s^2 \) (acceleration due to gravity),
- \( t \) is the time in seconds it takes to fall.
Rearranging the equation to solve for \( t \):
\[
t^2 = \frac{2h}{g}
\]
Now substitute the known values:
\[
t^2 = \frac{2 \times 1400}{9.81}
\]
Calculating the right side:
\[
t^2 = \frac{2800}{9.81} \approx 285.77
\]
Now taking the square root to find \( t \):
\[
t \approx \sqrt{285.77} \approx 16.91 \, s
\]
### Step 2: Calculate the Horizontal Distance
Now that we have the time of flight, we can calculate the horizontal distance the package will travel during this time using the horizontal speed of the jet.
The horizontal distance \( d \) can be calculated using the formula:
\[
d = v \cdot t
\]
Where:
- \( v = 91 \, m/s \) (horizontal speed of the jet),
- \( t \approx 16.91 \, s \) (time of flight).
Substituting the values:
\[
d = 91 \, m/s \cdot 16.91 \, s
\]
Calculating the distance:
\[
d \approx 91 \times 16.91 \approx 1534.81 \, m
\]
### Conclusion
The jet should drop the aid package approximately **1535 meters** (rounded to the nearest meter) away from the target cushion to ensure it lands correctly for the people to receive their aid undamaged.