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Question 335061: Solve the following by substitution and elimination:
On a football field during a game, Player A who runs the 40 yard dash in 4.9 seconds catches the ball at his own 30 yard line, then takes off running toward the goal line. Player B who runs the 40 yard dash in 4.5 seconds is standing on the 25 yard line when Player A catches the ball, he then pursues the ball.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! A runs the 40 yard dash in 4.9 seconds.
B runs the 40 yard dash in 4.5 seconds.
Speed of A is equal to 40 / 4.9 yards per second.
Speed of B is equal to 40 / 4.5 yards per second.
A catches the ball at his own 30 yard line.
B is on the 25 yard line when A catches the ball on the 30 yard line.
In order for B to catch A, he will have to run 5 more yards in the same time as B.
The basic formula is rate * time = distance.
The distance that B runs is represented by D.
The distance that A runs is represented by (D + 5) because A has to run 5 more yards than B in order to catch up to him.
The time that both runners are traveling is represented by T.
Since the formula is rate * time = distance, then:
Formula for A becomes 40/4.9 * T = D
Formula for B becomes 40/4.5 * T = D + 5
If we let a = 40/4.9 and we let b = 40/4.5, then:
Formula for A becomes a * T = D
Formula for B becomes b * T = D + 5
We need to solve both these equations simultaneously to get a common answer for both of them.
If we subtract formula for A from formula for B, then we get:
b*T - a*T = D+5 - D which becomes:
b*T - a*T = 5
Factor out the T to get:
T * (b-a) = 5
Divide both sides of this equation by (b-a) to get:
T = 5 / (b-a)
Since b = 40/4.5 and a = 40/4.9, this equation becomes:
T = 5 / (40/4.5 - 40/4.9)
We use our calculator to solve this to get T = 6.890625 seconds.
Since A and B are both traveling for the same amount of time, and because rate * time = distance, this means that:
B has traveled at a rate of 40/4.9 yards per second for 6.890625 seconds = 61.25 yards.
A has traveled at a rate of 40/4.9 yards per second for 6.890625 seconds = 56.25 yards.
A started at the 30 yard line and ran for 56.25 yards to get to the 86.25 yard line.
B started at the 25 yard line and ran for 61.25 yards to get to the 86.25 yard line in the same amount of time.
B catches up to A at the 86.25 yard line.
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