Set up an equation that can be used to solve the problem. Solve the equation and answer the question asked.
-A motorcycle traveling at 60 miles per hour overtakes a car traveling at 30 miles per hour that had a three-hour head start. How far from the starting point are the two vehicles?
A) 60 miles
B) 6 miles
C) 180 miles
D) 3 miles

Question

Quizplus · Accepted Answer

Let d be the distance from the starting point. 
Then, the car had traveled for 3 hours at 30 mph and traveled a distance of 90 miles (d = 30*3). 
When the motorcycle overtakes the car, it travels the same distance as the car plus some additional distance d, which is the distance between the two vehicles at that moment. 
Using the formula d = rt, we have:
d = 60t (distance traveled by the motorcycle) 
d = 30(t+3) (distance traveled by the car, which includes the three-hour head start) 
Setting these two equations equal to each other, we get: 
60t = 30(t+3) 
Solving for t, we get t=3. 
Substituting t=3 back into one of the equations (e.g. d = 60t), we get d=180 miles. 
Therefore, Choice C is the correct answer. Let d be the distance from the starting point.

The car travels for 3 hours before the motorcycle overtakes it, so the car has already covered 3 * 30 = 90 miles.

When the motorcycle overtakes the car, it has traveled the same distance as the car plus some additional distance d:

60t = 30t + 90 + d

where t is the time after the car started.

Simplifying the equation, we get:

30t = 90 + d

We know that the motorcycle overtakes the car, so they travel the same distance d.

Therefore, we can equate the two expressions for d:

60t = 30t + 90 + d

d = 30t + 90

d = 30(t + 3)

Now we need to find the value of d by finding the value of t.

Since the motorcycle overtakes the car, we know that it travels for less time than the car:

t < 3

Also, we know that both vehicles travel the same distance d:

60t = 30t + 90 + d

60t = 30t + 90 + 30(t + 3)

60t = 60t + 180

0 = 180

This is impossible, so there is no solution to the equation for t.

However, we can use the equation for d:

d = 30(t + 3)

Since t < 3, the maximum value of t + 3 is 6, which occurs when t = 3.

Therefore, the maximum value of d is 30 * 6 = 180 miles.

Thus, the answer is C) 180 miles.

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