To find the distance from the stronger source that minimizes the total illumination, we can follow these steps:
1. Let x be the distance from the stronger source, in feet. The distance from the weaker source is (10 - x).
2. The total received illumination at any point is given by:
Y = (3P / x^2) + (P / (10 - x)^2), where P is the radiant power of one of the light sources.
3. To find the minimum illumination, we need to find where dY/dx = 0. This means taking the derivative of Y with respect to x and solving for x.
4. Differentiating Y with respect to x:
dY/dx = (-6P / x^3) + (2P / (10 - x)^3)
5. Setting dY/dx = 0 and solving for x:
0 = (-6P / x^3) + (2P / (10 - x)^3)
Simplifying the equation, we can multiply both sides by x^3(10 - x)^3 to eliminate the fractions:
0 = (-6P)(10 - x)^3 + (2P)(x^3)
Expanding and rearranging:
0 = -6P(1000 - 300x + 30x^2 - x^3) + 2Px^3
Using the distributive property:
0 = -6000P + 1800Px - 180Px^2 + 6Px^3 + 2Px^3
Combining like terms:
0 = 8Px^3 - 180Px^2 + 1800Px - 6000P
Factoring out a common factor of 8P:
0 = 8P(x^3 - 22.5x^2 + 225x - 750)
Now, we have a cubic polynomial that we can solve for x.
6. To solve for x, we can use numerical methods or graphing calculators. By finding the roots of the cubic polynomial (x^3 - 22.5x^2 + 225x - 750 = 0), we can determine the values of x that minimize the total illumination.
Note: The specific values of P and the corresponding distances x will depend on the given radiant powers of the light sources.