During a hurricane evacuation from the east coast of Georgia, a family traveled 260 miles west. For part of the trip, they averaged 70 mph, but as the congestion got bad, they had to slow to 20 mph. If the total time of travel was 9 hours, how many miles did they travel at the reduced speed?
Show all work.
1--70t1 + 20t2 = 260
2--t1 + t2 = 9 or t1 = 9 - t2
3--Substitute (2) into (1) and solve for t2
4--Distance traveled at 20mph = 20t2
To find the number of miles traveled at the reduced speed, we need to first determine the distance traveled at the higher speed.
Let's denote the distance traveled at 70 mph as x miles.
Since the total distance traveled is 260 miles, the distance traveled at 20 mph can be found by subtracting x from the total distance:
Distance traveled at 20 mph = 260 miles - x miles.
Next, we can determine the time it took to travel the distance at 70 mph and the time it took to travel the distance at 20 mph.
Time traveled at 70 mph = Distance / Speed = x miles / 70 mph.
Time traveled at 20 mph = Distance / Speed = (260 miles - x miles) / 20 mph.
The total time of travel is given as 9 hours, so we can write the equation:
Time traveled at 70 mph + Time traveled at 20 mph = Total time of travel
x miles / 70 mph + (260 miles - x miles) / 20 mph = 9 hours.
Now, let's solve this equation for x:
(x / 70) + ((260 - x) / 20) = 9.
To simplify the equation, we can multiply both sides by the least common multiple (LCM) of 70 and 20, which is 140:
140 * (x / 70) + 140 * ((260 - x) / 20) = 140 * 9.
2x + 7(260 - x) = 1260.
2x + 1820 - 7x = 1260.
-5x = 1260 - 1820.
-5x = -560.
x = (-560) / (-5).
x = 112.
Therefore, the family traveled 112 miles at the higher speed of 70 mph.
Now, let's calculate the distance traveled at the reduced speed:
Distance traveled at 20 mph = 260 miles - x miles = 260 miles - 112 miles = 148 miles.
Therefore, the family traveled 148 miles at the reduced speed of 20 mph.