Certainly! Let's solve this problem step by step.
### Step 1: Define the Variables
Let's denote:
- [tex]\( v \)[/tex] as the plane's airspeed in mph.
- [tex]\( w \)[/tex] as the wind speed in mph.
### Step 2: Form the Equations
Using the given details:
1. When flying with the wind, the plane's effective speed is [tex]\( v + w \)[/tex]:
- Distance = 1890 miles
- Time = 3 hours
- Speed = Distance / Time
- Hence, [tex]\( 1890 = 3(v + w) \)[/tex]
2. When flying against the wind, the plane's effective speed is [tex]\( v - w \)[/tex]:
- Distance = 1890 miles
- Time = [tex]\( 3 \frac{3}{8} \)[/tex] hours (which equals [tex]\( 3.375 \)[/tex] hours)
- Speed = Distance / Time
- Hence, [tex]\( 1890 = 3.375(v - w) \)[/tex]
### Step 3: Solve the Equations
We have two equations now:
1. [tex]\( 1890 = 3(v + w) \)[/tex]
2. [tex]\( 1890 = 3.375(v - w) \)[/tex]
We can solve these simultaneously to find [tex]\( v \)[/tex] and [tex]\( w \)[/tex].
Let's rewrite the equations:
1. [tex]\( 1890 = 3v + 3w \)[/tex]
2. [tex]\( 1890 = 3.375v - 3.375w \)[/tex]
Divide both sides of each equation by the coefficients of [tex]\( v \)[/tex] and [tex]\( w \)[/tex]:
1. [tex]\( 630 = v + w \)[/tex]
2. [tex]\( 560 = v - w \)[/tex]
Now, solve these equations together by adding and subtracting:
- Adding the two equations:
[tex]\[
630 + 560 = (v + w) + (v - w) \implies 1190 = 2v \implies v = 595
\][/tex]
- Subtracting the second equation from the first:
[tex]\[
630 - 560 = (v + w) - (v - w) \implies 70 = 2w \implies w = 35
\][/tex]
### Step 4: Conclusion
- The plane's airspeed [tex]\( v \)[/tex] is [tex]\( 595 \)[/tex] mph.
- The wind speed [tex]\( w \)[/tex] is [tex]\( 35 \)[/tex] mph.
Hence, the wind speed is [tex]\( 35 \)[/tex] mph, and the plane's airspeed is [tex]\( 595 \)[/tex] mph.
