# Calculus optimization problem?

• Jan 10th 2013, 06:12 PM
achiu17
Calculus optimization problem?
The illumination from a bulb varies directly as the intensity of the light and Intensity varies inversely as the square of the distance from the source. Two bulbs are placed 54 feet apart. The intensity, Ia, of bulb A is 64cd, and the intensity, Ib, of bulb B is 125cd. At how many feet from bulb A along the line between the two bulbs is the total illumination the least?

I have no idea how to start this problem...I thought about using optimization but I'm not sure where to start
• Jan 10th 2013, 06:47 PM
MarkFL
Re: Calculus optimization problem?
I would orient my coordinate axis $x$ such that bulb A is at the origin 0 and bulb B is at 54, and let the observer be at $0. The intensity $I$ of the light received by the observer is then the sum of the light from the two bulbs, given by:

$I(x)=\frac{64}{x^2}+\frac{125}{(54-x)^2}$

Can you finish?
• Jan 10th 2013, 06:54 PM
achiu17
Re: Calculus optimization problem?
would i take the derivative of that function?
• Jan 10th 2013, 07:06 PM
MarkFL
Re: Calculus optimization problem?
Yes, you would equate the derivative to zero and solve for x to find the critical value. Then use one of the tests you have been taught (I recommend the second derivative test) to demonstrate you have a minimum.