Question

elias participated in a charity bike race. on the first half of the race, he biked at an average rate of 8 miles per hour to the turnaround point. on the second half of the race, he biked along the same path and biked at an average rate of 12 miles per hour. the total biking time was 5 hours. complete the equation that can be used to determine the time, t, elias spent biking to reach the turnaround point and find the total distance of the race. enter your answers in the boxes. equation: 8t = 12(□). time to reach the turnaround point: □ hours. total distance of the race: □ miles

Explanation:

Step1: Analyze the relationship between time and distance

Let \(t\) be the time to reach the turnaround point. The total time is \(5\) hours, so the time for the return trip is \((5 - t)\) hours.
The distance to the turnaround point is \(d_1=8t\) (using the formula \(d = rt\), where \(r = 8\) mph and \(t\) is time).
The distance of the return - trip is \(d_2 = 12(5 - t)\) (using the formula \(d=rt\), where \(r = 12\) mph and \(t_{return}=5 - t\)).
Since \(d_1=d_2\) (same path), the equation is \(8t=12(5 - t)\).

Step2: Solve the equation for \(t\)

Expand the right - hand side: \(8t=60-12t\).
Add \(12t\) to both sides: \(8t + 12t=60-12t+12t\), which gives \(20t=60\).
Divide both sides by \(20\): \(t=\frac{60}{20}=3\) hours.

Step3: Calculate the total distance of the race

The distance to the turnaround point \(d_1 = 8t\). Substitute \(t = 3\) into the formula: \(d_1=8\times3 = 24\) miles.
The total distance \(D=2d_1\) (round - trip), so \(D = 2\times24=48\) miles.

Answer:

• Equation: \(8t = 12(5 - t)\)
• Time to reach the turnaround point: \(3\) hours
• Total distance of the race: \(48\) miles