Question

Fiber-optic cables are used widely for internet wiring, data transmission, and surgeries. When light passes through a fiber-optic cable, its intensity decreases with the increase in the length of the cable. If 1500 lumens of light enters the cable, the intensity of light decreases by 3.4% per meter of the cable.
Can this situation be represented by a linear function?
Write a function f(x) to represent the intensity of light, in lumens, when it has passed through x meters of the cable.

Answer 1

Some scientists are trying to make a cable for which the intensity of light would decrease by 5 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 1500 lumens of light enter the cable

Answer 2

@KlOwNlOvE

Answer 3

a) The intensity of light after passing through 1 meter of cable is given by:
\[\large I _{1}=1500\times(1.000-0.034)=1500\times0.966\]
What will the intensity be after passing through 2 meters of cable?
\[\large I _{2}=?\]

Answer 4

1500*(1-0.034)^2

Answer 5

@kropot72

Answer 6

You are correct. And what will the intensity be after passing through x meters of cable?

Answer 7

I don't know, how would i do that?

Answer 8

can it be represented by a linear function or not?

Answer 9

\[\large I _{1}=1500\times0.966\]
\[\large I _{2}=1500\times0.966^{2}\]
\[\large I _{3}=1500\times0.966^{3}\]
\[\large I _{4}=1500\times0.966^{4}\]
\[\large I _{x}=1500\times......?\]

Answer 10

oh, I got it, the intensity will be increasing at the same value each meter, which can be explained by finding the rate of change.

Answer 11

Not really.
Calculating the intensity after 1, 2 and 3 meters gives:
\[\large I _{1}=1449\]
\[\large I _{2}=1399.734\]
\[\large I _{3}=1352.143\]
\[\large I _{1}-I _{2}=49.266\]
\[\large I _{2}-I _{3}=47.591\]
The required equation will not be linear.

Answer 12

The required equation can be seen by looking at the pattern that I posted previously. It is as follows:
\[\large f(x)=1500\times0.966^{x}\]

Answer 13

what is the .966 from??

Answer 14

b) In the second case the intensity decreases by a fixed amount per unit length. Therefore the equation will be linear, the reason being there will be no higher powers of the variable higher than unity.

Answer 15

hmm but wouldn't it be 1.034, because f(x)=p(1+r)^x

Answer 16

but if its decreasing I guess it would be .966

Answer 17

"what is the .966 from??"
The intensity in part a) decreases by 3.4% per meter. Therefore the intensity after 1 meter will be:
\[\large I _{1}=1500\times(1.000-0.034)=1500\times0.966\]

Answer 18

k gotcha, so the function is f(x)=1500(1-.034)^x

Answer 19

Yes, you are correct for part a)
The equation for part b) the equation is:
\[\large f(x)=1500-5x\]

Answer 20

So that would be for the scientist one?

Answer 21

And that is linear

Answer 22

@kropot72

Answer 23

Correct.

Answer 24

thanks!!