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 congestion got bad, they had to slow to 20 mph. If the total time of travel was 9 hours, how many miles did they drive at the reduced speed?
My brain is frozen and has shutdown because of all the ways I have tried to solve this problem. Please help and Thank you
Rt=d
r=70
y.= r-50
X+y=260
70x=y(x-50)
T=9 hrs
70x=20(x-50)
if they traveled x hours at 20 mi/hr, then they went (260-20x) miles at 70 mi/hr
Since time = distance/speed,
(260-20x)/70 + x = 9
x = 7.4
check: they traveled
1.6 hr * 70 mi/hr = 112 mi
7.4 hr * 20 mi/hr = 148 mi
Total: 260 mi
To solve this problem, let's break it down step by step:
1. Let's assign some variables to the unknowns in the problem. Let x represent the time (in hours) that the family traveled at 70 mph. Then y represents the time (in hours) that they traveled at 20 mph.
2. Now we can set up two equations based on the given information. The first equation represents the total distance traveled: x + y = 260 miles.
3. The second equation represents the total time taken for the trip: x(70) + y(20) = 9 hours. Notice that we multiply the time traveled at each respective speed by the speed itself to calculate the distance traveled at that speed.
4. Simplify the second equation by multiplying the speeds by their respective times:
70x + 20y = 9
5. Rearrange the first equation to solve for x:
x = 260 - y
6. Substitute the value of x in the second equation:
70(260 - y) + 20y = 9
7. Distribute 70 to the terms inside the parentheses:
18,200 - 70y + 20y = 9
8. Combine like terms:
-50y = 9 - 18,200
-50y = -18,191
9. Divide both sides of the equation by -50 to solve for y:
y = -18,191 / -50
y = 363.82
10. Since the distance traveled cannot be negative, we discard the negative solution. Therefore, y = 363.82 is not a valid solution.
11. Now, substitute the value of y back into the first equation to solve for x:
x + 363.82 = 260
x = 260 - 363.82
x = -103.82
12. Again, we discard the negative solution, so x = -103.82 is not valid.
13. As there is no valid solution in this case, we can conclude that the information given in the problem is not accurate or does not reflect a possible scenario.
In summary, we were unable to find a valid solution to this problem based on the given information. It's possible that there may have been an error or inconsistency in the problem statement.