Problem 763

The intensity I of light form a light bulb varies inversely as the square of the distance d from the light bulb. Suppose I is 490 w/m2 when the distance is 4 m. How much farther would it be to a point where the intensity is 250 w/m2?




First, familiarize yourself with the problem. You told that the intensity, I of light from the light bulb varies inversely as the square of the distance d to the light bulb.

This means the intensity and distance can be related by the equation

I =\frac{k}{4^{2}}

Solve to get k = 7840.

Substitute into the formula for intensity,

I =\frac{7840}{d^{2}}.

Next , transfer the problem. You want to find how much farther away you would to be for the intensity to be 250 w/m2. If the additional distance is x, write an equation that relates this distance to the intensity of 250 w/m2.

The distance d = 4 +x.

Thus, the equation can be written to relate this distance to the intensity.

250 = \frac{7840}{(4+x)^{2}}

To simplify, multiply through by a common denominator.

250*(4+x)2 = 7840

Divide both sides by 250 to isolate the quadratic square.


Take the square root of both sides. Since 4+x is supposed to represent a distance, only the positive square root needs to be considered.

4 + x = \sqrt{\frac{7840}{250}}

4 + x = \frac{28}{5}

Solve this for x by subtracting 4 from both sides.

x = \frac{28}{5}-4

x = \frac{28}{5} - \frac{20}{5}

x = \frac{8}{5}

Always check your answer. Substitute x = \frac{8}{5} and find that the intensity is equal to 250 w/m, just as expected.

Finally, convert the fraction to a decimal. You would have to be an additional 1.6 meter(s) away to reduce the intensity to 250 w/m2.



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