19. Can someone help me please, with this College Algebra

Question

An autoist made a trip of 450 miles in 8 hours. Before noon he averaged 60 mph, and after noon he averaged 50 mph. at what time did he begin his trip and when did he end it ? 

Answer 1

Hi Shane,

 

Let's begin by writing algebraic expressions to represent our two unknowns:

    time at 60 mph = x

    time at 50 mph = 8 - x

 

Next, let's write an equation to express the sum of the distances traveled at each time and speed and solve it for our two unknowns:

    (60)(x) + (50)(8-x) = 450

    60x + 400 - 50x = 450

    10x + 400 = 450

    10x = 450 - 400

    10x = 50

    x = 50/10

    x = 5 hours (at 60 mph)

    8 - x = 8 - 5

    8 - x = 3 hours (at 50 mph)

 

We can verify our two time answers by plugging them back into our equation to see that the sum of the distances traveled at each time and speed is equal to the given trip distance:

    (60)(x) + 50(8 - x) = 450

    60(5) + 50(3) = 450

    300 + 150 = 450

    450 miles = 450 miles (distances are equal) 

 

Finally, we can use our two time answers to determine the start and end times of the trip.  We are told that the autoist traveled at the faster speed (60 mph) before noon and at the slower speed (50 mph) after noon:

    noon - 5 hours (at 60 mph) = 7:00 AM (start time)

    noon + 3 hours (at 50 mph) = 3:00 PM (end time)

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.

Answer 2

The formula we need here is  distance = rate x time

 

The question is about time so let's use t as our variable, and let t represent the number of hours he traveled in the morning (before noon). Since he drove a total of 8 hours (and didn't even stop for lunch!), then the time he drove after noon is expressed by  8-t

 

Make sure you understand that before we go on. That is so the two times together make 8 hours.

 

For t hours he drove 60 mph, so the distance he covered was  60t miles

 

For (8-t) hours he drove 50 mph, so the distance he covered was

50(8-t) miles.

 

Stop again to make sure you know what we are doing.

 

Since his total distance was 450 miles, we have the equation

 

   60t + 50(8-t) = 450    (morning miles + afternoon miles = total miles)

 

Now we have the equation to solve. Use the distributive property to remove the parentheses.

 

   60t + 400 - 50t = 450

 

Combine like terms

 

   10t + 400 = 450

 

Subtract 400 from both sides

 

  10t = 50

 

Divide by 10 to get    t = 5 hours.

 

We aren't finished yet, because the question was when he started and when he finished.

 

He started 5 hours before noon, which is 7:00 AM. He finished 8 hours later (or 3 hours after noon) at 3:00 PM