It takes a plane 40 minutes longer to fly from Boston to Los Angeles at 525 miles per hour than it does to return at 600 miles per hour. How far apart are the cities?
Let T be the shorter of the two times (in hours), and D be the distance
600 T = D
525 (T + 2/3) = D = 525 T + 350
600 T = 525 T + 350
75 T = 350
T = 4 2/3 hours
D = 600 T = 2800 miles
Well, it seems like Boston and Los Angeles have some serious relationship issues. First, Boston flies away at 525 miles per hour and Los Angeles gets jealous, so it takes 40 minutes longer. Then, when Los Angeles tries to come back at 600 miles per hour, Boston is all about "I don't have time for this" and it takes less time. Typical.
Anyway, let's solve this. Let's assume the distance between Boston and Los Angeles is "x" miles. We'll start with the equation:
x/525 = (x/600) - (40/60)
Now, let's do some math magic. Simplifying the right side of the equation, we get:
x/525 = (x/600) - (2/3)
To get rid of the denominators, let's multiply everything by 525 and 600:
600x = 525x - 525(2/3)
Now, simplify that (I'm not a math tutor, by the way):
600x = 525x - 350
Subtract 525x from both sides:
75x = -350
Divide both sides by 75:
x = -350/75
Now, we don't want any negativity in our answer, so let's make it positive:
x = -4.6666667
Well, that's a problem. Did Boston and Los Angeles move closer together or something? They seem quite clingy, but I doubt they're in the negative distance zone.
I think there might be a mistake somewhere, because Boston and Los Angeles are definitely not stuck inside each other. Although, that would explain Boston driving LA crazy.
To find the distance between the cities, we need to use the formula:
distance = speed × time
Let's assume the distance between Boston and Los Angeles is "d" miles.
When traveling at a speed of 525 miles per hour, the time taken to fly from Boston to Los Angeles is given by:
time = distance / speed
time = d / 525
When traveling at a speed of 600 miles per hour, the time taken to return from Los Angeles to Boston is given by:
time = distance / speed
time = d / 600
According to the problem, it takes the plane 40 minutes longer to fly from Boston to Los Angeles than to return. Since there are 60 minutes in an hour, we can write this as:
d / 525 = d / 600 + 40/60
Simplifying the equation:
d / 525 = d / 600 + 2/3
To solve this equation, we get rid of the fractions by multiplying through by the LCD (Least Common Denominator), which is 31500:
31500 * (d / 525) = 31500 * (d / 600) + 31500 * (2/3)
Simplifying further:
60d = 52.5d + 21000
Collecting like terms and solving for d:
7.5d = 21000
d = 21000 / 7.5
Therefore, the distance between Boston and Los Angeles is:
d = 2800 miles
So, the cities are 2800 miles apart.
To solve this problem, we can utilize the formula:
Distance = Speed × Time
Let's assume the distance between Boston and Los Angeles is represented by "d" miles.
Given that the plane takes 40 minutes longer to fly from Boston to Los Angeles than it does to return, we can determine the time it takes to fly from Los Angeles to Boston by adding 40 minutes (or 40/60 = 2/3 hours) to the time it takes to fly from Boston to Los Angeles.
Now we can set up equations for both directions:
For the flight from Boston to Los Angeles:
d = 525 × t1
For the flight from Los Angeles to Boston:
d = 600 × (t1 - 2/3)
Since we have two equations with the same "d" value, we can solve for "d" by setting the two expressions for "d" equal to each other:
525 × t1 = 600 × (t1 - 2/3).
Let's solve this equation step by step:
525 × t1 = 600 × t1 - 400
Isolating t1, we subtract 525t1 from both sides:
0 = 600 × t1 - 525 × t1 - 400
Simplifying further:
0 = 75 × t1 - 400
Adding 400 to both sides:
400 = 75 × t1
Dividing both sides by 75:
400/75 = t1
t1 ≈ 5.33 hours
Now we can substitute the value of t1 into one of the original equations (d = 525 × t1) to find the distance:
d = 525 × 5.33
d ≈ 2790 miles
Therefore, the distance between Boston and Los Angeles is approximately 2790 miles.