Solve each problem. Speed of a PlaneMary Lynn left by plane to visit her mother in Louisiana, 420 km away. Fifteen minutes later, her mother left to meet her at the airport. She drove the 20 km to the airport at 40 km per hr, arriving just as the plane taxied in. What was the speed of the plane?

Verified step by step guidance

1. Define the variables: Let \(s\) represent the speed of the plane in km/hr.
2. Express the time taken by the mother to drive to the airport: Since she drove 20 km at 40 km/hr, her time is \(t_{mother} = \frac{20}{40} = 0.5\) hours.
3. Express the time taken by the plane to travel 420 km: The plane's time is \(t_{plane} = \frac{420}{s}\) hours.
4. Set up the relationship between the times: The mother left 15 minutes (which is 0.25 hours) after the plane, but they arrived at the same time. So, the plane's travel time is 0.25 hours longer than the mother's travel time, which gives the equation \(t_{plane} = t_{mother} + 0.25\).
5. Substitute the expressions for \(t_{plane}\) and \(t_{mother}\) into the equation: \(\frac{420}{s} = 0.5 + 0.25\). Then solve this equation for \(s\) to find the speed of the plane.

Key Concepts

Relative Motion and Meeting Point

This concept involves understanding how two moving objects starting at different times and places can meet at the same point. Here, the plane and the mother travel towards the airport, and their travel times and distances relate to when and where they meet.

Distance, Speed, and Time Relationship

The fundamental formula connecting distance, speed, and time is distance = speed × time. This relationship allows solving for an unknown variable when the other two are known, which is essential for finding the plane's speed in this problem.

Unit Conversion and Time Adjustment

Since the mother leaves 15 minutes after the plane, converting minutes to hours (15 minutes = 0.25 hours) is necessary to align time units. Properly adjusting and comparing travel times ensures accurate calculation of speeds and distances.