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If a plane can travel 470 miles per hour with the wind and 390 miles per hour against the wind, find the speed of the wind and the speed of the plane in still air.

What is the speed of the wind?
___ mph

Sagot :

GinnyAnswer
To solve for the speed of the wind and the speed of the plane in still air, we'll denote the following variables:

- [tex]\( p \)[/tex]: the speed of the plane in still air.
- [tex]\( w \)[/tex]: the speed of the wind.

Given two scenarios:
1. The plane travels 470 miles per hour with the wind, which means [tex]\( p + w = 470 \)[/tex].
2. The plane travels 390 miles per hour against the wind, which means [tex]\( p - w = 390 \)[/tex].

These two equations represent the relationship between the plane’s speed in still air, the wind speed, and the observed speeds with and against the wind:

1. [tex]\( p + w = 470 \)[/tex]
2. [tex]\( p - w = 390 \)[/tex]

To find the values of [tex]\( p \)[/tex] and [tex]\( w \)[/tex], let's solve these equations step-by-step:

### Step 1: Add the Two Equations

We start by adding the two equations together:

[tex]\[ (p + w) + (p - w) = 470 + 390 \][/tex]

Simplifying:

[tex]\[ p + w + p - w = 860 \][/tex]

This reduces to:

[tex]\[ 2p = 860 \][/tex]

Dividing both sides by 2:

[tex]\[ p = 430 \][/tex]

So, the speed of the plane in still air, [tex]\( p \)[/tex], is 430 miles per hour.

### Step 2: Substitute [tex]\( p \)[/tex] into One of the Original Equations

Next, we substitute [tex]\( p \)[/tex] into one of the original equations to solve for [tex]\( w \)[/tex]. We'll use [tex]\( p + w = 470 \)[/tex]:

[tex]\[ 430 + w = 470 \][/tex]

Subtracting 430 from both sides:

[tex]\[ w = 470 - 430 \][/tex]

[tex]\[ w = 40 \][/tex]

So, the speed of the wind, [tex]\( w \)[/tex], is 40 miles per hour.

### Conclusion

The speed of the wind is 40 mph.
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