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 four times as strong as the other, are placed 11 ft apart, how far away from the stronger light source should an object be placed on the line between the two sources so as to receive the least illumination?
Let the object be placed x feet from the weaker source. Then
ks/x^2 = k(4s)/(11-x)^2
or
4x^2 = (11-x)^2
4x^2 = 121 - 22x + x^2
3x^2 + 22x -121 = 0
(x+11)(3x-11) = 0
BRIGHT_____x__WEAK
8'4" -- 3'8"
To find the distance from the stronger light source where the object should be placed to receive the least illumination, we need to apply the given information.
Let's assume that the distance from the object to the stronger light source is x feet. According to the given problem, the distance from the object to the weaker light source would be (11 - x) feet since the two light sources are 11 ft apart.
Now, let's refer to the strengths of the light sources. We are told that the stronger light source is four times as strong as the weaker one. Let's represent the strength of the weaker light source as w, so the strength of the stronger light source would be 4w.
According to the given information, illumination is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source.
Thus, the illumination from the stronger light source would be (4w/x^2), and the illumination from the weaker light source would be (w/(11-x)^2).
To find the total illumination, we add these two illuminations together:
Illumination = (4w/x^2) + (w/(11-x)^2)
To find the point where the object receives the least illumination, we need to minimize this expression.
To minimize a function, we can take the derivative of the function with respect to x, set it equal to 0, and solve for x. However, before we do that, let's simplify the expression:
Illumination = (4w/x^2) + (w/(11-x)^2)
= (4w/x^2) + (w/(11-x))^2
= (4w(x - 11)^2 + wx^2) / (x^2(11 - x)^2)
Now, let's find the derivative of this expression with respect to x:
d(Illumination)/dx = (d/dx) [(4w(x - 11)^2 + wx^2) / (x^2(11 - x)^2)]
= ((8w(x - 11) + 2wx)(x^2(11 - x)^2) - (4w(x - 11)^2 + wx^2)(2x(11 - x))) / (x^4(11 - x)^4)
Now, we can set this derivative equal to 0 and solve for x:
((8w(x - 11) + 2wx)(x^2(11 - x)^2) - (4w(x - 11)^2 + wx^2)(2x(11 - x))) / (x^4(11 - x)^4) = 0
Simplifying this equation may be quite complex due to the squared terms involved. However, numerically solving this equation will give us the value of x at which the object should be placed to receive the least illumination.
To solve this problem, we need to find the distance from the stronger light source that will result in the least illumination on the object.
Let's call the distance from the stronger light source x (in ft). Also, let's assume the illumination at this distance is I.
According to the given information, the illumination of an object is directly proportional to the strength of the source and inversely proportional to the square of the distance from the source:
I1 = k * (S1/x^2) -- Equation 1 (for the stronger light source)
I2 = k * (S2/(11-x)^2) -- Equation 2 (for the weaker light source)
where I1 and I2 are the illumination from the stronger and weaker light sources, S1 and S2 are the strengths of the stronger and weaker light sources, and k is a constant of proportionality.
Given that the stronger light source is four times as strong as the weaker light source:
S1 = 4 * S2
Substituting this into Equation 1 and Equation 2, we have:
I1 = k * (4 * S2 / x^2) -- Equation 3
I2 = k * (S2 / (11-x)^2) -- Equation 4
Since we want to find the distance that results in the least illumination, we need to minimize the sum of I1 and I2:
Total Illumination = I1 + I2
Substituting the values from Equations 3 and 4, we get:
Total Illumination = k * (4 * S2 / x^2) + k * (S2 / (11-x)^2)
Total Illumination = k * (4 * S2 / x^2 + S2 / (11-x)^2)
To find the minimum illumination, we need to find the value of x that minimizes the total illumination. To do that, we can take the derivative of the Total Illumination with respect to x and set it equal to zero:
d(Total Illumination) / dx = 0
Let's differentiate the Total Illumination equation:
d(Total Illumination) / dx = k * ([-8 * S2 / x^3] + [2 * S2 / (11-x)^3])
0 = -8 * S2 / x^3 + 2 * S2 / (11-x)^3
Multiplying through by x^3 and (11-x)^3 to eliminate the denominators:
0 = -8 * S2 * (11-x)^3 + 2 * S2 * x^3
Simplifying further:
-8 * (1331 - 363x + 33x^2 - x^3) + 2 * x^3 = 0
-10688 + 2904x - 264x^2 + 8x^3 + 2x^3 = 0
10x^3 - 264x^2 + 2904x - 10688 = 0
Unfortunately, there is no simple algebraic solution for this cubic equation. We need to use numerical methods or a calculator to solve for x.