The illumination of an object by a light source is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source. If two light sources, one three times as strong as the other, are placed 10 ft apart, how far in ft from the stronger source should an object be placed on the line between the sources so as to receive the least TOTAL illumination (the sum of the illumination from each source)?
Let x be the distance from the stronger source, in feet. The distance from the weaker source is therefore (10 - x). Let the total radiant power of the brighter souce be 3P and the other equal to P.
The received illumination is
Y = 3P/x^2 + P/(10-x)^2
Differentiate that to find where dY/dx = 0. Mimimum illumination will be there. You won't need to know P; it will cancel out.
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.