Solve each problem. Speed of a Plane Mary 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?

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Define variables: Let \(s\) represent the speed of the plane in km/hr.

Express the time taken by the plane to travel 420 km as \(\frac{420}{s}\) hours.

Since Mary Lynn's mother left 15 minutes (which is \(\frac{15}{60} = \frac{1}{4}\) hour) after the plane departed, her travel time is \(\left(\frac{420}{s} - \frac{1}{4}\right)\) hours.

Set up the equation for the mother's trip: distance = speed \(\times\) time, so \(20 = 40 \times \left(\frac{420}{s} - \frac{1}{4}\right)\).

Solve the equation \(20 = 40 \left(\frac{420}{s} - \frac{1}{4}\right)\) for \(s\) to find the speed of the plane.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Relative Motion and Time

Understanding how two moving objects start at different times and travel towards the same point is essential. Here, the plane departs first, and the mother leaves 15 minutes later, so their travel times differ. Calculating the time each takes helps relate their speeds and distances.

Distance-Speed-Time Relationship

The fundamental formula distance = speed × time connects these three variables. Knowing any two allows calculation of the third. In this problem, distances and speeds are given or sought, so applying this formula correctly is key to finding the plane's speed.

Unit Conversion and Consistency

Ensuring consistent units, especially for time, is crucial. The mother leaves 15 minutes later, so converting minutes to hours (15 minutes = 0.25 hours) aligns with speeds given in km/h. This consistency avoids errors in calculations involving speed and time.

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