Let's say you are standing on top of a mountain looking down at the valley below, where you can see cars driving on a road. You stretch out your arm, use your thumb to "sight" a car, and find that the car appears to be as long as your thumb is wide. Estimate how far you are from the car. Explain your method clearly. (You will need to make an assumption to solve this problem. Make a realistic assumption and make it clear what your assumption is.)
To estimate the distance from the car, we can use a method called the Thumb Rule or Thumb Method. This method involves making a realistic assumption and using the concept of similar triangles.
Assumption: The assumption we need to make is that the average width of a car is approximately 1.8 meters (or 6 feet).
Method:
1. Hold your arm straight out in front of you, fully extended, and stretch your thumb to align with the car you are observing.
2. Observe how many times the car's width fits between your thumb's width. Let's say it appears to fit 5 times.
3. Now, measure the width of your thumb using a ruler in inches or centimeters. Let's say your thumb's width is 3 centimeters (cm).
4. Convert the car's width to the same unit you used to measure your thumb's width. In this case, 1.8 meters is equal to 180 centimeters (cm).
5. Calculate the estimated distance using the concept of similar triangles:
Distance to the car = (Thumb's width in cm) * (Actual car's distance) / (Car's width in cm)
In this case: Distance to the car = 3 cm * (actual car's distance) / 180 cm
Simplifying the equation, the car's distance ≈ 60 * Thumb's Width
Therefore, based on this method and the measurements given, the estimated distance from the car is approximately 180 meters (60 * 3 cm = 180 cm), assuming the average car width is 1.8 meters. This method provides a rough estimate and may not be completely accurate due to variations in car sizes and the accuracy of your observation.