Anna is driving from Champaign to Indianapolis on I-74. She passes the Prospect Ave. exit at noon and maintains a constant speed of 75 mph for the entire trip. Chuck is driving in the opposite direction. He passes the Brownsburg, IN exit at 12:30pm and maintains a constant speed of 65 mph all the way to Champaign. Assume that the Brownsburg and Prospect exits are 105 miles apart, and that the road is straight.
How far from the Prospect Ave. exit do Anna and Chuck pass each other?
From 12:00 to 12:30, Anna goes 37.5 miles
That means that there remains 67.5 miles to Brownsburg.
The two cars' combined speed is 140 mi/hr, so it takes 67.5/140 = 0.48 hours to meet.
I expect you can finish up now, eh?
Why are we adding the two car's speeds?
If it is to find the time at which the two cars meet at, wouldn't we be using the [x = (1/2)(V-V(i))*t]?
To find out how far from the Prospect Ave. exit Anna and Chuck pass each other, we need to calculate the total distance traveled by both of them.
Since Anna started at noon and maintained a constant speed of 75 mph, the time taken by her to reach the point of intersection is given by the formula:
Time = Distance / Speed
Let x be the distance from the Prospect Ave. exit where they meet.
The total distance traveled by Anna is x miles.
Now, let's find out the time taken by Anna to travel this distance:
Time taken by Anna = x miles / 75 mph
Similarly, Chuck started at 12:30pm and maintained a constant speed of 65 mph. Let's assume the distance traveled by Chuck to the point of intersection is (105 - x) miles.
The time taken by Chuck to travel this distance is given by:
Time taken by Chuck = (105 - x) miles / 65 mph
Since they start at different times, the total time taken by Anna and Chuck to reach the point of intersection should be the same.
So, we have the equation:
Time taken by Anna = Time taken by Chuck
x miles / 75 mph = (105 - x) miles / 65 mph
Now, let's solve this equation to find the value of x.
To find out how far from the Prospect Ave. exit Anna and Chuck pass each other, we need to determine their distances traveled from their respective starting points.
Let's start by calculating the time it took for Chuck to reach the point of intersection:
Chuck started driving at 12:30 pm and reached the point of intersection when Anna passed Prospect Ave. at noon. So, the time difference is 30 minutes or 0.5 hours.
We know that Chuck's speed is 65 mph, so by using the formula Distance = Speed × Time, we can determine the distance Chuck traveled in that time:
Distance traveled by Chuck = Speed × Time
= 65 mph × 0.5 hours
= 32.5 miles
Now, let's calculate the distance from the Prospect Ave. exit where Anna and Chuck passed each other:
The Prospect Ave. exit is 105 miles away from the Brownsburg exit.
Since Anna and Chuck are driving towards each other, their distances traveled sum up to the total distance between the Prospect Ave. and Brownsburg exits.
Distance traveled by Anna + Distance traveled by Chuck = Total distance
= Approaching distance + Departing distance
= 105 miles
As Anna started at the Prospect Ave. exit and drove at a constant speed of 75 mph until they met Chuck, we can calculate the distance traveled by Anna using the formula Distance = Speed × Time:
Distance traveled by Anna = Speed × Time
= 75 mph × Time
Since Anna and Chuck met at the point of intersection, their travel times are the same. Therefore, the distance traveled by Anna is equal to the total distance minus the distance traveled by Chuck:
Distance traveled by Anna = Total distance - Distance traveled by Chuck
= 105 miles - 32.5 miles
= 72.5 miles
So, Anna and Chuck pass each other 72.5 miles from the Prospect Ave. exit.