Question

Please help me!!!!!
Fiber-optic cables are used widely to increase the speed and accuracy of data transmission. When light passes through a fiber-optic cable, its intensity decreases with the increase in the length of the cable. If 1700 lumens of light enters the cable, the intensity of light decreases by 1.9% per meter of the cable.

Answer 1

Part A: Can this situation be represented by a linear function? Justify your answer.
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.
Part C: Some scientists are trying to make a cable for which the intensity of light would decrease by 2 lumens per unit length of the cable. Can this situation be represented by a linear function? Justify your answer and write the appropriate function to represent this situation if 1700 lumens of light enter the cable.

Answer 2

Part A: By the statement of the problem, I would say yes.

Answer 3

I mean, the intensity decreases with the length, so it is possible to relate distance and intensity by means of an afine function or linear function of the type y=m*x+n

Answer 4

Part B: You should find the slope (negative in this case) that is the decreasing rate of intensity. n is equal to the original intensity, 1700. Once you have done that, you will have the function y=mx+n

Answer 5

Part C: the same, but you have to recalculate the slope of the function.

Answer 6

Can you help me with those parts?

Answer 7

Yes, what part do you need?

Answer 8

I am just super confused by this question. Can you help me with Part A, B and C by clearing everything up

Answer 9

Ok, let's see. For part A, you can argue that the problem says there is a proportional relationship between intensity and length. As there is a proportional relationship (in this case, the higher the distance the lower the intensity) then you know intensity is a linear function of the distance.

Answer 10

Part B: Now we need a function that describes mathematically the process. We know that a linear function can be represented by the formula
\[y=m x+n\] where in our case, y represents the intensity and x the length.

Answer 11

We need those m and n. To calculate them, we use the problem. The problem says the initial intensity is 1700 lumens, so\[n=1700\]. Also, it says that the decreasing rate is 1.9%, so we can say,\[m=-\frac{1.9}{100}=-0.019\]. Then, our inal function should be of the form,
\[y=-0.019x+1700\]

Answer 12

You can rewrite it in this way,\[f(x)=-0.019x+1700\]

Answer 13

Part C: To justify this question use the same argument as in part A. To calculate the function you only to recalculate the slope m, but the problem says it must be m=-2. So,\[f(x)=-2x+1700\]. And that's all.