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 10 times as strong as the other, are placed 9 ft apart, how many feet from the brighter source should an object be placed on the line between the sources so as to receive the least illumination?
let the light be situated x ft from one light and
9-x from the other.
Let the power be P, where P is a constant
let I be the total illumination.
I = P/x^2 + 10P/(9-x)^2
= Px^-2 + 10P(9-x)^-2
dI/dx = -2Px^-3 - 10P(9-x)^-3 (-1)
= 0 for a min of I
10P/(9-x)^3 = P/x^2
divide by P , which is a constant
10/(9-x)^3 = 1/x^3
10x^3 = (9-x)^3
10^(1/3) x = 9-x
10^(1/3) x + x = 9
x(10^(1/3) - 1 = 9
x = 7.796 ft
It should be placed appr 7.8 ft away from the stronger light
or 1.2 ft from the weaker light
To determine the position where the object should be placed to receive the least illumination, we need to find the point on the line between the two sources where the sum of the illuminations from each source is minimized.
Let:
- x be the distance from the brighter source to the object,
- y be the distance from the weaker source to the object.
According to the given information, the illuminations from each source can be expressed as follows:
For the brighter source, the illumination is directly proportional to its strength:
I₁ = S₁
For the weaker source, the illumination is inversely proportional to the square of the distance:
I₂ = S₂ / (y²)
Now, let's analyze the situation when the object is placed at a distance x from the brighter source. In this case, the distance from the weaker source to the object would be 9 - x.
So, the total illumination can be calculated by summing the illuminations from each source:
I_total = I₁ + I₂
= S₁ + (S₂ / (9 - x)²)
To find the minimum illumination, we need to find the value of x that minimizes I_total.
Now, let's substitute the given values into the equation:
Since S₁ = 10 * S₂, we can rewrite the equation as:
I_total = 10 * S₂ + (S₂ / (9 - x)²)
To find the minimum of this function, we can take the derivative with respect to x and set it equal to zero:
d(I_total)/dx = 0
To simplify this, let's solve it step by step:
d(I_total)/dx = 0
10 * (-2) * S₂ / (9 - x)³ + 2 * S₂ / (9 - x)³ = 0
-20 * S₂ / (9 - x)³ + 2 * S₂ / (9 - x)³ = 0
-18 * S₂ / (9 - x)³ = 0
Since S₂ ≠ 0, we can ignore this equation. So, the derivative is zero only when the denominator equals zero:
9 - x = 0
Solving this equation, we find:
x = 9
Therefore, the object should be placed 9 feet from the brighter source on the line between the two sources to receive the least illumination.
To find the position where the object receives the least illumination, we need to determine the distance from the brighter light source. Let's go step by step:
1. Let's assume that the distance from the brighter light source to the object is represented by 'x' feet. Since the two light sources are placed 9 ft apart, the distance from the dimmer light source to the object would be (9 - x) ft.
2. According to the given information, the illumination of an object is inversely proportional to the square of the distance from the source. Therefore, the illumination (I) can be expressed as:
I1/I2 = (D2^2)/(D1^2)
Where I1 and I2 are the illuminations produced by the brighter and dimmer light sources, and D1 and D2 are the distances from the object to the brighter and dimmer light sources, respectively.
3. Since the brighter light source is 10 times stronger than the dimmer light source, we can write:
I1/I2 = 10/1
Therefore, (D2^2)/(D1^2) = 10/1
4. Using the given distances, we substitute the values into the equation:
((9 - x)^2)/(x^2) = 10/1
5. We can simplify the equation by cross-multiplying:
(9 - x)^2 = 10 * x^2
Expanding and rearranging the equation:
81 - 18x + x^2 = 10x^2
9x^2 + 18x - 81 = 0
6. Now we have a quadratic equation that can be solved to find the value of 'x'. We can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
Plugging in the values:
x = (-18 ± sqrt(18^2 - 4 * 9 * -81))/(2 * 9)
Simplifying:
x = (-18 ± sqrt(324 + 2916))/(18)
x = (-18 ± sqrt(3240))/(18)
x = (-18 ± 180)/(18)
x = (-18 + 180)/(18) or x = (-18 - 180)/(18)
x = 162/18 or x = -198/18
x = 9 or x = -11
Since the distance can't be negative, the only valid solution is x = 9.
7. Therefore, the object should be placed 9 feet from the brighter light source on the line between the sources to receive the least illumination.