Allison drove home at 66 mph, but her brother Austin, who left at the same time, could drive at only 48 mph. When Allison arrived, Austin still had 54 miles to go. How far did Allison drive?
They both drove for the same time of t hrs
distance covered by Allison = 60t
distance covered by Austin = 48t
60t - 48t = 54
12t = 54
t = 54/12 = 4.5
So they both drove for 4.5 hours
and Allison went 270 miles
(Austin went 4.5(48) or 216 miles, which is 54 miles less than Allison. )
To determine how far Allison drove, we can use the formula: distance = rate × time.
Let's denote the distance that Allison drove as D. We know that Allison drove at a rate of 66 mph and her brother, Austin, drove at a rate of 48 mph. We will also need to find the time both of them spent driving.
Let's denote the time Allison drove as T. Since Austin left at the same time, we can denote his time as T as well.
For Allison, distance = rate × time, so D = 66T.
For Austin, distance = rate × time, so 54 = 48T.
We now have two equations based on the given information:
D = 66T -- equation (1)
54 = 48T -- equation (2)
To find the distance Allison drove, we need to solve these equations simultaneously.
Let's start by solving equation (2) for T:
48T = 54
T = 54 / 48
T = 1.125 hours
Now that we know the time T, we can substitute it back into equation (1) to find D:
D = 66T
D = 66 * 1.125
D = 74.25 miles
Therefore, Allison drove approximately 74.25 miles.
To find the distance that Allison drove, we need to calculate the total distance traveled by Austin and then subtract the distance he still had to go when Allison arrived.
Let's first calculate the distance traveled by Austin:
Austin's speed = 48 mph
Time taken by Austin = Distance / Speed
Distance traveled by Austin = Speed × Time = 48 mph × Time (let's assume the time taken by Austin is 't')
We know that the time taken by Austin is the same as Allison because they left at the same time. So, the distance traveled by Austin is the same as the distance traveled by Allison when she arrived.
Now, let's calculate the distance traveled by Austin:
Distance traveled by Austin = Distance traveled by Allison when she arrived = Austin's speed × Time = 48 mph × t
We also know that Austin had 54 miles to go when Allison arrived. So, the distance traveled by Austin when Allison arrived is:
Distance traveled by Austin when Allison arrived = Total distance traveled by Austin - Distance still to go = (48 mph × t) - 54 miles
Now, we can set up an equation since both distances are equal when Austin arrived at his destination:
Distance traveled by Austin when Allison arrived = Distance traveled by Allison when she arrived
(48 mph × t) - 54 miles = Distance traveled by Allison when she arrived
Since Allison drove at a speed of 66 mph and the time taken by both is the same, the distance traveled by Allison when she arrived can be calculated as follows:
Distance traveled by Allison when she arrived = Allison's speed × Time = 66 mph × t
Equating the two equations, we have:
(48 mph × t) - 54 miles = 66 mph × t
Now, let's solve for t:
48t - 54 = 66t
-54 = 66t - 48t
-54 = 18t
t = -54 / 18
t = 3
Therefore, Austin took 3 hours to reach his destination, and we can substitute this value of t to find the distance traveled by Allison when she arrived:
Distance traveled by Allison when she arrived = 66 mph × 3 hours = 198 miles
Thus, Allison drove a distance of 198 miles.