Final answer:
Piper's initial walking speed to Kenzie's house is 5 km/h, found by setting up an equation based on total distance and time, and then solving the resulting quadratic equation.
Step-by-step explanation:
To solve this rational problem, let's denote Piper's initial walking speed to Kenzie's house as x km/h. For the return trip, she walked 3 km/h slower, which would be (x - 3) km/h. The distance to Kenzie's house and back is the same, 10 km each way.
Using the formula time = distance / speed, we can write two equations for the time it takes to walk to Kenzie's house and back:
- Time to Kenzie's: 10 km / x km/h
- Time back home: 10 km / (x - 3) km/h
According to the problem, the total walking time for the round trip was 7 hours:
10/x + 10/(x - 3) = 7
To solve this equation, we need to clear the denominators by multiplying through by the common denominator, x(x - 3), which gives us:
10(x - 3) + 10x = 7x(x - 3)
This simplifies to:
10x - 30 + 10x = 7x^2 - 21x
Combining like terms, we get a quadratic equation:
7x^2 - 41x + 30 = 0
Applying the quadratic formula x = [-b ± √(b^2 - 4ac)] / (2a) (where a=7, b=-41, and c=30) yields two potential solutions for x. Only one of these will make sense in the context of the problem (a positive speed that fits within the 7-hour time frame).
After calculating, we find that the reasonable, whole number solution for Piper's initial walking speed to Kenzie's house is 5 km/h.