Question 102409: If Tory averaged 52 mph on state highways and 62 mph on interstate highways on her 450 mile trip and her driving time was 7 hours and 45 minutes, how long did she travel on interstate highways and how many miles did she travel on state highways? Write an equation for this problem that relates time and distance, where t=the time Tory traveled on state highways.
Answer by mjgulinello(10)   (Show Source):
You can put this solution on YOUR website! First of all, we know the total distance (D) Tory traveled was 450 miles. All of this travel was on either state highways or interstate highways. The other thing we know is that the total travel time was 7 hours 45 minutes, or 7.75 hours.
Given that t is the time traveled on state highways, we can find the distance traveled on state highways (Ds) using the following equation:
Distance = speed * time
Ds = 52t
We also know that all the time Tory was not on state highways, she was on interstate highways. Therefore, the time spent on interstate highways (Ti) is
Ti = 7.75 - t
So the distance she traveled on interstate highways (Di) can be expressed as
Di = 62 * Ti or
Di = 62(7.75 – t)
Because we know Tory only traveled on state highways and interstate highways, we can write the total distance Tory traveled as
D = Ds + Di
or if we plug in the equations above:
D = 52t + 62(7.75 – t)
Remember we also know that the total distance was 450 miles, so we can plug in 450 for D, giving us one variable to solve for.
450 = 52t + 480.5 – 62t
First combine like terms on the right side.
450 = 480.5 – 10t
Then add 10t to each side of the equation.
10t + 450 = 480.5
Subtract 450 from each side of the equation.
10t = 30.5
Divide each side of the equation by 10.
t = 3.05
So the time Tory traveled on state highways was 3.05 hours, or 3 hours 3 minutes.
Now we can go back and plug in t.
Ti = 7.75 - t
Ti = 7.75 - 3.05
Ti = 4.7
Tory spent 4.7 hours, or 4 hours 42 minutes on interstate highways.
Ds = 52t
Ds = 52 * 3.05
Ds = 158.6
Tory traveled 158.6 miles on state highways.
Di = 62 * Ti
Di = 62 * 4.7
Di = 291.4
Tory traveled 291.4 miles on interstate highways.
Now we have solved all the components of the problem, so let's go back and make sure our solution works. Remember the distances should add up to 450 miles.
D = Ds + Di
450 = 158.6 + 291.4
450 = 450
So our solution checks out. That's it for this problem.
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