Average Speed

Question

A family drove 1980 miles to their vacation lodge. Because of increased traffic density, their average speed on the return trip was decreased by 6 miles per hour and the trip took 2.5 hours longer. Determine their average speed on the way to the lodge.

Answer 1

Hi Sarah, this is an exercise using the formula: D(istance) = R(ate) * T(ime). We know the distance to the lodge and the distance home from the lodge is the same.

 

Let x represent the speed on the trip to the lodge and t represent the time it takes to get to the lodge. Using the given information we will get two distance equations.

 

Equation 1, to the lodge:

 

D_l = x * t

 

Equation 2, from the lodge:

 

D_h = (x - 6) * (t + 2.5)

 

The distance is the same in both directions, therefore, D_l = D_h.

 

This becomes:

 

x * t = (x - 6) * (t + 2.5)

 

x * t = x * t + 2.5x - 6t -15

 

Combine terms:

 

2.5x - 6t - 15 = 0

 

Solve for x:

 

x = (6t + 15) / 2.5

 

Using equation 1, substitute for x:

 

(6t + 15) / 2.5 * t = 1980

 

Solve for t using the quadratic equation:

 

t = (-15 ± √((-15²) - 4 * 6 * - 1980)) / 12

 

t = (-15 - 345) / 12, or t = -30. We can discard this answer since time cannot be negative in this equation.

 

t = (-15 + 345) / 12, or t = 27.5. The trip to the lodge took 27.5 hours.

 

To find the rate of speed to the logde, divide 1980 by 27.5, or the rate to the lodge was 72 miles per hour.

 

Check:

 

Substitute for x and t in equation 2:

 

1980 = (72 - 6) * (27.5 + 2.5)

 

1980 = 66 * 30

 

1980 = 1980, values check.

 

Answer:

 

The rate of speed to the lodge was 72 miles per hour and it took 27.5 hours to get there.

 

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