A plane traveling from Chicago to Boston with a tailwind of 10 miles per hour takes 4 hours. The return trip in the same wind, which is now a headwind, takes the same plane 4 hours and 9 minutes. According to this problem, about how far is it from Chicago to Boston?
since distance = speed * time,
d = (s+10)*4
d = (s-10)*(4 9/60)
so,
4s+40 = (4 9/60)s - 10*(4 9/60)
s = 1630/3
So, d = 6640/3 mi
check:
4 * (1630/3 + 10) = 6640/3
(4 9/60) * (1630/3 - 10) = 6640/3
To determine the distance between Chicago and Boston, we need to use the formula:
Distance = Rate × Time
For the outbound trip with a tailwind:
Rate = Plane's speed + Tailwind's speed = Plane's speed + 10 miles per hour
Time = 4 hours
For the return trip with a headwind:
Rate = Plane's speed - Headwind's speed = Plane's speed - 10 miles per hour
Time = 4 hours + 9 minutes = 4.15 hours
Since the distance traveled for the outbound trip is the same as the return trip, we can set up the following equation:
(Plane's speed + 10) × 4 = (Plane's speed - 10) × 4.15
Let's solve for Plane's speed:
4(Plane's speed + 10) = 4.15(Plane's speed - 10)
4Plane's speed + 40 = 4.15Plane's speed - 41.5
0.15Plane's speed = 81.5
Plane's speed = 81.5 / 0.15
Plane's speed = 543.33 miles per hour (approx.)
Now, we can calculate the distance:
Distance = Rate × Time
Distance = (543.33 + 10) × 4
Distance = 553.33 × 4
Distance = 2213.32 miles (approx.)
Therefore, it is approximately 2,213.32 miles from Chicago to Boston.
To find the distance from Chicago to Boston, we can first calculate the average speed of the plane in still air. Let's call this speed "x" miles per hour.
In the first scenario, the plane is traveling with a tailwind of 10 miles per hour. This means that its effective speed (the speed of the plane relative to the ground) is increased by 10 mph. So, the speed of the plane going from Chicago to Boston is (x + 10) mph.
Using the formula Speed = Distance / Time, we can determine the distance traveled by the plane in 4 hours: Distance = Speed * Time. Applying it to the first scenario, we have:
Distance = (x + 10) * 4
In the second scenario, the plane is now facing a headwind, which reduces its effective speed by 10 mph. So, the speed of the plane returning from Boston to Chicago is (x - 10) mph.
Again, using the Speed = Distance / Time formula, we can calculate the distance traveled in 4 hours and 9 minutes (or 4.15 hours):
Distance = (x - 10) * 4.15
Since the distance from Chicago to Boston is the same in both directions, we can set the two equations equal to each other:
(x + 10) * 4 = (x - 10) * 4.15
Now, we can solve this equation to find the value of x, which represents the average speed of the plane in still air.
Distributing and simplifying the equation, we get:
4x + 40 = 4.15x - 41.5
Combining like terms, we have:
0.15x = 81.5
Dividing both sides by 0.15, we find:
x = 543.33
So, the average speed of the plane in still air is approximately 543.33 miles per hour.
Now, we can find the distance between Chicago and Boston by substituting this value back into either of the original equations. Let's use the first equation for simplicity:
Distance = (x + 10) * 4
Distance = (543.33 + 10) * 4
Distance = 553.33 * 4
Distance = 2213.32 miles
Therefore, the distance from Chicago to Boston is approximately 2213.32 miles.