The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of three meters. Find the intensity level at eight meters.
So with this kind of question, we need to pull out the key information. So what we’ve got is that the intensity varies inversely with the square of the distance. So we actually use that here with our proportionality sign. So we’ve got the intensity is inversely proportional to the square of the distance and that’s shown as inversely proportional as one over the distance squared.
Okay, but this isn’t very much use. What we need here is an equal sign. And to actually enable us to create a formula, what we need is the proportionality constant or 𝑘. So we can say that 𝐼, our intensity, is equal to 𝑘 over 𝑑 squared. So great, we’ve actually now got a formula that we can use.
So what’s the next step? Well, the next step is to actually find 𝑘. So whenever you get a problem like this, the first thing to do is actually find our proportionality constant. And to do that, we’re actually gonna use some values we know from the question. Well, we know from the question that the intensity of a light bulb is 0.08 foot-candles at a distance of three meters. So we’re gonna say 𝐼 is equal 0.08 and 𝑑 is equal to three.
So now, what we can do is actually substitute this in to our formula to actually help us find out what 𝑘 is. And when we do that, we get 0.08 is equal to 𝑘 over three squared. So therefore, we get 0.08 is equal to 𝑘 over nine and that’s cause three squared is nine. So therefore, if we multiply each side of the equation by nine, we’re gonna get 𝑘 is equal to 0.72.
And obviously, we can use the calculator to work that out. But if we’re actually doing that by hand, then the way we do it is nine multiplied by eight gives us 72. But then, we can see that it’s nine multiplied by 0.08. So if there are two numbers after the decimal point in our first calculation, that has to be in our final answer. So we get 0.72.
Okay, great, now, we can actually get into the final part of the question and solve our problem and that is to find the intensity level at eight meters. But before we can get on and do that, what we’re gonna do is substitute our value of 𝑘 back into our formula to give us a new formula. And we’ve got that now. And it is 𝐼, so the intensity, is equal to 0.72, our value for 𝑘, divided by 𝑑 squared, so the distance squared.
Well, we know from the question that the intensity is what we want to find and the distance is eight meters. So again, what we’re gonna do is actually substitute this into our formula. So we’re gonna get that 𝐼 is equal to 0.72 over eight squared, which gives us 𝐼 is equal to 0.72 over 64. And that’s because eight squared is equal to 64.
So therefore, we can say that if the intensity of light measured in foot-candles varies inversely with the square of the distance from the light source, then if we have the intensity of a light bulb is 0.08 at a distance of three meters, the intensity at eight meters is going to be equal to 0.01125 foot-candles.