Fiber-optic cables are used widely by broadcast and cable companies. When light passes through a fiber-optic cable, its intensity decreases with the increase in the length of the cable. If 1000 lumens of light enters the cable, the intensity of light decreases by 1.2% per meter of the cable.
Part A: Can this situation be represented by an exponential function? Justify your answer. (2 points)
Part B: Write a function f(x) to represent the intensity of light, in lumens, when it has passed through x meters of the cable. (4 points)
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Part C: Some scientists are trying to make a cable for which the intensity of light would decrease by 8 lumens per unit length of the cable. Can this situation be represented by an exponential function? Justify your answer and write the appropriate function to represent this situation if 1000 lumens of light enter the cable. (4 points)
I only need help with Part C
So that means you need to construct an exponential function which makes a straight line graph, doesn't it?
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1000,1000-8=992, 992-8=984, 984-8=976, etc. That looks like \(y = -8x+1000\) to me. Is it possible to make an exponential that does that?
I want to say no but the x variable throws me off..
your instinct is correct. Only way you get a straight line out of an exponential is if you take a logarithm.
in my equation, \(x\) was the distance along the fiber from the starting point, and \(y\) was the intensity.
So it can be exponential?
no, it cannot. An exponential is not a straight line, because a straight line would be either of the form \[y=k\]horizontal line\[x=k\]vertical line\[y=mx+b\]something in between, with slope \(m\) and \(y\)-intercept \(b\)
and an exponential will be of the form \[y = kb^x\]
where \(k\) is a constant, and \(b\) is a constant representing the multiplicative factor.
for example, the equation showing the intensity if it drops 1.2% with each meter of fiber:
\[y = 1000(1-0.012)^x = 1000(0.988)^x\]