Let's solve the problem step-by-step.
1. Define Variables:
- Let the speed of the plane in still air be [tex]\( v \)[/tex] mph.
- The speed of the wind is given as 25 mph.
2. Set Up Equations:
- When the plane is flying with the wind, the effective speed is [tex]\( v + 25 \)[/tex] mph.
- When the plane is flying against the wind, the effective speed is [tex]\( v - 25 \)[/tex] mph.
3. Time Calculation:
- The distance covered with the wind is 440 miles.
- The distance covered against the wind is 340 miles.
- The time taken to cover a distance can be calculated using the formula [tex]\( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)[/tex].
4. Time Equations:
- Time taken to travel 440 miles with the wind: [tex]\( \frac{440}{v + 25} \)[/tex]
- Time taken to travel 340 miles against the wind: [tex]\( \frac{340}{v - 25} \)[/tex]
5. Set up the Equation for Equal Times:
- Since the times are the same, we can equate the two expressions: [tex]\( \frac{440}{v + 25} = \frac{340}{v - 25} \)[/tex]
6. Solve for [tex]\( v \)[/tex]:
- Cross-multiply to solve for [tex]\( v \)[/tex]:
[tex]\[
440 \cdot (v - 25) = 340 \cdot (v + 25)
\][/tex]
- Expand both sides:
[tex]\[
440v - 11000 = 340v + 8500
\][/tex]
- Combine like terms by moving all [tex]\( v \)[/tex] terms to one side and constant terms to the other side:
[tex]\[
440v - 340v = 11000 + 8500
\][/tex]
[tex]\[
100v = 19500
\][/tex]
- Solve for [tex]\( v \)[/tex]:
[tex]\[
v = \frac{19500}{100}
\][/tex]
[tex]\[
v = 195
\][/tex]
Therefore, the speed of the plane in still air is [tex]\( 195 \)[/tex] mph.
