A cyclist travels 5 miles each way on a bike path. Travel in one direction is with a 4 mile per hour wind and travel back is against a 4 mile per hour wind. Part A: Write and simplify an equation for the amount of time t in hours it will take the cyclist to make the trip. Part B: What is the cyclist's total travel time, if she travels an average of 6 miles per hour without the wind?

a. 10v/(v² - 16) b. 3 hours

Step-by-step explanation:

a. Write and simplify an equation for the amount of time t in hours it will take the cyclist to make the trip.

Let v be the speed of the cyclist without the wind. The speed of the cyclist with the 4 mile per hour wind in the direction of the wind is v' = v + 4. Since the distance travelled is 5 miles, and time, t = distance/speed, the time it takes to make this trip is t' = 5/(v + 4)

Let v be the speed of the cyclist. The speed of the cyclist with the 4 mile per hour wind against the direction of the wind is v" = v - 4. Since the distance travelled is 5 miles, and time, t = distance/speed, the time it takes to make this trip is t" = 5/(v - 4).

So, the total travel time, T = t' + t"

= 5/(v + 4) + 5/(v - 4)

= [5(v - 4) + 5(v + 4)]/[(v + 4)(v - 4)]

= [5v - 20 + 5v + 20)]/(v² - 4²)

= 10v/(v² - 16)

b. What is the cyclist's total travel time, if she travels an average of 6 miles per hour without the wind?

If v = 6 miles per hour, then

T = 10v/(v² - 16)

= 10(6)/(6² - 16)

= 60/(36 - 16)

= 60/20

= 3 hours

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step-by-step explanation: