I drove to the beach at a rate of 40 miles per hour. If I had driven at a rate of 35 miles per hour instead, then I would have arrived 45 minutes later. How many miles did I drive?
Let's call the distance you drove "d" and the time it took you to drive that distance at 40 mph "t".
Using the formula distance = rate × time:
d = 40t
We also know that if you had driven at 35 mph instead, you would have arrived 45 minutes (or 0.75 hours) later. So the time it would have taken you to drive that same distance "d" at 35 mph would be:
t + 0.75
Using the formula again, but this time with a rate of 35:
d = 35(t + 0.75)
Now we have two equations for "d", which we can set equal to each other:
40t = 35(t + 0.75)
40t = 35t + 26.25
5t = 26.25
t = 5.25
So it took you 5.25 hours to drive to the beach at 40 mph.
Now we can use either of the two equations we set up earlier to find "d". Let's use the first one:
d = 40t
d = 40(5.25)
d = 210
So you drove 210 miles to get to the beach.
To solve this problem, we can use the equation Distance = Rate × Time.
Let's assume the distance you drove is D miles.
At a rate of 40 miles per hour, the time it took is given by the equation:
D = 40 × T1
Similarly, at a rate of 35 miles per hour, the time it would have taken is given by the equation:
D = 35 × T2
We also know that T2 is 45 minutes (or 0.75 hours) longer than T1.
Therefore, we can set up the equation:
T2 = T1 + 0.75
Let's substitute the second equation into the first equation:
D = 35 × (T1 + 0.75)
Now, since we have two equations for D, we can equate them:
40 × T1 = 35 × (T1 + 0.75)
Simplifying the equation:
40 × T1 = 35 × T1 + 26.25
Combining like terms:
5 × T1 = 26.25
Dividing both sides by 5:
T1 = 5.25
Now, we can find the distance by substituting T1 into the first equation:
D = 40 × 5.25
Calculating the distance:
D = 210 miles
Therefore, you drove 210 miles to the beach.