Question

every weekend, you ride your bicycle on a forest preserve path. the path is 24 miles long and ends at a waterfall, at which point you relax and then make the trip back to the starting point. one weekend, you find that in the same time it takes you to travel to the waterfall, you are only able to return 20 miles. your average speed going to the waterfall is 2 miles per hour faster than the return trip. use the formula time traveled = \\(\frac{distance traveled}{average velocity}\\) to find your average speed going to the waterfall. the average speed going to the waterfall is miles per hour.

Explanation:

Step1: Let the average speed of return trip

Let the average speed of the return trip be $x$ miles per hour. Then the average speed going to the waterfall is $(x + 2)$ miles per hour.

Step2: Calculate time for each part

The time taken to go to the waterfall $T_1=\frac{24}{x + 2}$ (using $T=\frac{d}{v}$, where $d = 24$ miles and $v=x + 2$). The time taken for the return - trip $T_2=\frac{20}{x}$ (using $T=\frac{d}{v}$, where $d = 20$ miles and $v = x$).

Step3: Set up the equation

Since the time taken to go to the waterfall is the same as the time taken for the return - trip, we have the equation $\frac{24}{x + 2}=\frac{20}{x}$.

Step4: Cross - multiply

Cross - multiplying gives $24x=20(x + 2)$.

Step5: Expand and solve for $x$

Expand the right - hand side: $24x=20x+40$. Subtract $20x$ from both sides: $24x-20x=40$, so $4x = 40$, and $x = 10$.

Step6: Find the speed going to the waterfall

The average speed going to the waterfall is $x + 2$. Substitute $x = 10$ into $x + 2$, we get $10+2=12$ miles per hour.

Answer:

12