Help with a Wave Intensity Problem

[Smx_Drummerboy]

[SOLVED] Help with a Wave Intensity Problem

Hello all, here is a problem that relates to Wave Motion, more specifically Wave Intensity I believe:

The figure shows two observers 20m apart on a line that connects them and a spherical light source.
Figure: (The best I know to do)

(Person A)<----------20m----------->(Person B)<----------?m---------->(Light Source)

If the observer nearer the source measures a light intensity 45 % greater than the other observer, how far is the nearer observer from the source?

Relevant Equations:
I = [ latex ] \frac{P}{(4pi(r^2))} [ /latex ]

My initial thought was to state the following:
Ia = .45Ib and da = x +20 and db = x

Then plug in all the values given, the P's cancel, and I thought I'd be left with an answer. I got 41m, apparently that isn't right. I've been working on this problem for nearly an hour and a half.

Any suggestions?

 

[Nate810]

Solution:We can use the intensity equation given to solve for the distance.Ia = .45Ib [latex]\frac{P}{(4\pi (d_a^2))}[/latex] = .45[latex]\frac{P}{(4 \pi (d_b^2))}[/latex] Cross multiplying and simplifying, we get:[latex]d_a^2 = 0.45d_b^2+20^2[/latex] Solving for d_a, we get:[latex]d_a = \sqrt{0.45d_b^2+20^2}[/latex] Substituting db = 20m and solving, we get:[latex]d_a = \sqrt{0.45(20^2)+20^2}[/latex] [latex]d_a = 22.4m[/latex] Therefore, the observer nearer to the source is 22.4m away from it.

 

[ogb p]

Hello, thank you for reaching out for help with this wave intensity problem. It seems that you have made a good start by setting up the relevant equations and variables. However, there are a few things to consider in order to solve this problem correctly.

Firstly, it is important to note that the equation for wave intensity is actually I = P/(4πr^2), where P is the power of the source and r is the distance from the source. The equation you have written includes a term for the surface area of a sphere, which is not necessary for this problem.

Secondly, it is not necessary to introduce new variables for the distances between the observers and the source. We can simply use the given information that the observers are 20m apart and the intensity at one observer is 45% greater than the other.

So, let's start by writing the equation for Person A (closer to the source):
Ia = P/(4πra^2)

And for Person B:
Ib = P/(4πrb^2)

Now, we can use the given information to set up the following equation:
Ia = 1.45Ib

Substituting the equations for Ia and Ib, we get:
P/(4πra^2) = 1.45P/(4πrb^2)

We can cancel out the P's and rearrange the equation to solve for ra:
1.45rb^2 = ra^2

Taking the square root of both sides, we get:
ra = 1.2rb

Since we know that the distance between the observers is 20m, we can set up another equation:
rb = ra - 20

Substituting this into the previous equation, we get:
ra = 1.2(ra-20)

Solving for ra, we get:
ra = 40m

Therefore, the distance between Person A and the light source is 40m.

I hope this helps you to solve the problem correctly. If you have any further questions, please don't hesitate to ask. Good luck!