Help Needed: Shadow Length Calculation Problem

[tandoorichicken]

Its been a while since I did something like this:

A man 6 ft tall walks at the rate of 5 ft/sec toward a street light that is 16 ft above the ground. At what rate is the length of his shadow changing when he is 10 ft from the base of the light?

And I forgot how.
Please help. Thx.

 

[himanshu121]

https://www.physicsforums.com/showthread.php?s=&threadid=10792

The above link will help this is the same kind of Problem

 

[M98Ranger]

Sure, I'd be happy to help with this problem. First, let's review the given information. We have a man who is 6 ft tall walking towards a street light that is 16 ft above the ground. We also know that he is walking at a rate of 5 ft/sec. The question is asking us to find the rate at which the length of his shadow is changing when he is 10 ft from the base of the light.

To solve this problem, we can use the concept of similar triangles. Since the man and his shadow are both standing on the ground, we can create a right triangle with the man's height (6 ft) as the vertical side, the distance from the base of the light to the man (10 ft) as the horizontal side, and the length of the shadow as the hypotenuse.

Now, we can use the Pythagorean theorem to find the length of the shadow. The equation would be: (length of shadow)^2 = (6 ft)^2 + (10 ft)^2. Solving for the length of the shadow, we get approximately 11.66 ft.

To find the rate at which the length of the shadow is changing, we can use the chain rule from calculus. The equation would be: (rate of change of shadow) = (rate of change of distance from base) * (rate of change of shadow length with respect to distance).

The first rate of change is given to us in the problem as 5 ft/sec. The second rate of change can be found by taking the derivative of the equation we found earlier for the length of the shadow. d/dx (length of shadow) = (1/2)*(1/sqrt((6 ft)^2 + x^2)) * 2x. Plugging in x=10 ft, we get a rate of change of approximately 0.57 ft/ft.

Finally, we can multiply these two rates together to get the final answer of approximately 2.85 ft/sec. This means that the length of the man's shadow is increasing at a rate of 2.85 ft every second when he is 10 ft away from the base of the light.

I hope this helps you understand the problem and how to solve it. Let me know if you have any further questions. Good luck!