# Finding rate of change of shadow problem.

• ugeous
In summary, the conversation involves using similar triangles to solve for the derivative of x and y in a given equation, with one of the derivatives already known. The solution is found by plugging in the given value and solving for the unknown derivative.
ugeous
Question: A 1.85 m tall man is walking toward a 12 m tall street light at night at a rate of 2.2 m/s. How fast is the length of his shadow changing when he is 12 m from the street light?

so, using similar triangles, i got that (x+y)/12 = x/1.85. I can rearrange this into x= or y=, but then I don't see how i can find the derivative of it (ex. x = (1.85x+1.85y)/12 -> finding derivative will eliminate x and y). Any help will be greatly appreciated.

Thanks!

The derivative of that is (dx/dt+dy/dt)/12=dx/dt/1.85. The problem statement tells you what one of those derivatives is. Which one and what's it's value?

Oh I think I got it. So just to make sure: I solve for dx/dt plugging in -2.2 for dy/dt correct?

ugeous said:
Oh I think I got it. So just to make sure: I solve for dx/dt plugging in -2.2 for dy/dt correct?

Exactly.

Thanks a lot Dick!

## 1. What is the rate of change of a shadow?

The rate of change of a shadow refers to how fast the length of the shadow is changing. This can be calculated by comparing the length of the shadow at different times, or by using the position of the sun and the object casting the shadow.

## 2. How do you find the rate of change of a shadow?

To find the rate of change of a shadow, you need to measure the length of the shadow at two different times. Then, you can use the formula "change in length of shadow/change in time" to calculate the rate of change. Alternatively, you can use the tangent of the angle between the sun and the object to find the rate of change.

## 3. Why is it important to find the rate of change of a shadow?

Finding the rate of change of a shadow can help determine the position and movement of the sun, which can be useful for navigation, determining time, and predicting astronomical events. It can also be used for practical purposes such as designing buildings and solar panels to optimize their exposure to sunlight.

## 4. What factors affect the rate of change of a shadow?

The rate of change of a shadow is affected by the position of the sun, the position and size of the object casting the shadow, and the distance between the object and the surface on which the shadow is cast. It can also be affected by atmospheric conditions such as clouds or fog.

## 5. Can the rate of change of a shadow be negative?

Yes, the rate of change of a shadow can be negative if the length of the shadow is decreasing over time. This can happen when the sun is setting or when the angle of the sun and the object is decreasing. In this case, the shadow is getting shorter and the rate of change is negative.

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