How Fast Is Shadow Moving


Related rate problems in mathematics can produce very counter-intuitive results. The classic example is where some object is moving away from a light  source and a shadow is being cast. One can ask how fast is the tip of the shadow moving or at what rate the shadow is lengthening?

Even though a related rate problem is technically a calculus problem, algebra and trig usually play a larger role in solving the problem.

From the picture you can see a 6 foot tall person walking away from a 15 foot light pole at a speed of 5 feet per second. So we can ask what is the speed of the tip of the shadow that he is casting? What follows is a step by step solution to the problem.

Related Rate Speed of Shadow from a Light Source


Each Step of Related Rate Shadow Problem

 Some of the key steps are to first write down the two equal proportions that result from the ratios of corresponding sides of similar triangles are proportional AND then cross multiply the equal fractions AND then differentiate the resulting simplified equation with respect to t. Do not forget that all variables in related rate problems are understood to be functions of time.

Also note that there was only one step where any calculus was done. The rest of the work was algebraic and numerical in character.


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