A family drives 1,260 miles to a vacation lodge. On the return trip, it takes the family
2 1\2 hours longer, traveling at an average speed that is 7 miles per hour slower. Determine the average speed (in mph) on the way to the lodge.
To determine the average speed on the way to the lodge, we can use the formula: speed = distance / time.
Let's assign variables to the unknowns:
- Let r be the average speed on the way to the lodge (in mph).
- Let t1 be the time taken to drive to the lodge.
- Let t2 be the time taken on the return trip.
Given information:
- The distance traveled to the lodge is 1,260 miles.
- On the return trip, the time taken is 2 1/2 hours longer than the time taken on the way to the lodge.
- On the return trip, the average speed is 7 mph slower.
Based on the given information, we can write the following equations:
Equation 1: r = 1260 / t1
Equation 2: r - 7 = 1260 / (t1 + 2.5)
To solve this system of equations, we can substitute Equation 1 into Equation 2:
(1260 / t1) - 7 = 1260 / (t1 + 2.5)
To simplify the equation further, we multiply through by t1(t1 + 2.5):
1260(t1 + 2.5) - 7t1(t1 + 2.5) = 1260t1
Simplifying the equation gives:
1260t1 + 3150 - 7t1^2 - 17.5t1 = 1260t1
Moving all terms to one side:
7t1^2 + 17.5t1 - 3150 = 0
We can now solve for t1 using the quadratic formula:
t1 = (-17.5 ± sqrt(17.5^2 - 4(7)(-3150))) / (2(7))
After calculating, we get two values for t1: -35 and 21.428.
Since time cannot be negative in this context, we discard the negative value of t1. Therefore, t1 = 21.428 hours.
Finally, we can substitute t1 into Equation 1 to find the average speed on the way to the lodge:
r = 1260 / 21.428 ≈ 58.82 mph.
Therefore, the average speed on the way to the lodge is approximately 58.82 mph.