Posted on April 22, 2022 (Updated on July 31, 2025)

How do you find shadow related rates?

Space & Navigation

Chasing Shadows: Making Sense of Related Rates

Okay, so you’re diving into calculus, and suddenly you’re faced with “related rates” problems. Sounds intimidating, right? But trust me, once you get the hang of it, it’s actually pretty cool stuff. And among these problems, the ones about shadows? They’re a classic. Think about it: a person walks away from a lamppost, and their shadow stretches out longer and longer. The question is, how fast is that shadow growing? That’s the kind of thing we’re talking about.

At its heart, related rates is all about connections. Imagine two things changing at the same time – like, say, the radius of a balloon as you inflate it, and the volume of the balloon itself. As one changes, so does the other. If you can figure out the equation that links them, you can figure out how their rates of change are related too. Simple as that!

Now, shadow problems are a perfect example. You’ve got someone strolling along, casting a shadow that’s constantly morphing. The trick is to see how the person’s movement affects the shadow’s length, and how both of those things affect the tip of the shadow. Are we trying to figure out how fast the shadow’s growing, or how fast the end of the shadow is moving? It makes a difference!

So, how do you actually solve these things? Here’s my tried-and-true method:

  • Draw a Picture, Seriously: I know, it sounds basic, but sketching out the scene is so important. Get a visual! Draw the person, the lamppost, the shadow. Label everything. Trust me, it helps.

  • Name Those Players: Give everything a name – a variable, that is.

    • x: How far the person is from the light.
    • y (or s): The length of the shadow itself.
    • L (or z): The total distance from the light to the tip of the shadow (that’s x + y).
    • Don’t forget the constants: The height of the person and the lamppost don’t change.

  • Find the Link: This is the clever bit. You need to find an equation that connects all these variables. Usually, it involves similar triangles – remember those from geometry? The big triangle formed by the lamppost and the entire shadow is similar to the smaller triangle formed by the person and their shadow. This gives you a proportion you can work with. Something like this:

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