Question

A motorist makes a trip of 180 miles. For the first 90 miles, she drives at a constant speed of 30 mph. At what constant speed must she drive the remaining distance if her average speed for the total trip is to be 40 mph?

Answer 1

D = 1.5 miles

Average Speed = Total Distance / Total Time

In this case, the total distance is 180 miles, and the average speed is 40 mph. We need to find the total time for the trip. Let's denote the remaining distance as D (in miles) and the constant speed for the remaining distance as V (in mph).

So, for the first 90 miles, the time taken is:

Time1 = Distance1 / Speed1 = 90 miles / 30 mph = 3 hours

Now, for the remaining distance D, the time taken is:

Time2 = Distance2 / Speed2 = D miles / V mph

The total time for the entire trip is the sum of Time1 and Time2:

Total Time = Time1 + Time2 = 3 hours + (D miles / V mph)

We know that the average speed is 40 mph, and the total distance is 180 miles. Therefore:

Average Speed = Total Distance / Total Time

40 mph = 180 miles / (3 hours + D miles / V mph)

Now, we need to solve for D:

40 mph = 180 miles / (3 hours + D miles / V mph)

Now, cross-multiply to eliminate the denominator:

40 mph * (3 hours + D miles / V mph) = 180 miles

Next, distribute the 40 mph:

120 hours + 40D mph = 180 miles

Now, isolate the term with D:

40D mph = 180 miles - 120 miles

40D mph = 60 miles

Now, divide both sides by 40 mph to solve for D:

D = (60 miles) / (40 mph)

D = 1.5 miles