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.
Part A: Can this situation be represented by a linear 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)
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. (4 points)
well, do part B and you will have the answer to part A
100 - 1.9 = 98.1 percent
so every time we go a meter, we multiply by .981
after x meters
f(x) = 1700 * .981^x (Part B answer)
which is exponential, not linear. (part A answer) It does not lose a constant "amount" per meter but a constant "fraction".
in part C, a constant "amount" is lost per meter so this is indeed linear (if impossible)
I = 1700 - 2 x
f(x) =
Part A: To determine whether this situation can be represented by a linear function, we need to understand the characteristics of a linear function. A linear function is one that has a constant rate of change, meaning that the change in the dependent variable (intensity of light in this case) is constant for each unit change in the independent variable (length of the cable in this case).
In this situation, the intensity of light is decreasing by 1.9% per meter of the cable. Since the rate of decrease is not constant but dependent on the length of the cable, it cannot be represented by a linear function. Therefore, the answer is no.
Part B: We can represent the intensity of light using an exponential function, since the decrease is not linear. Let's define f(x) as the intensity of light, in lumens, when it has passed through x meters of the cable.
Since the intensity decreases by 1.9% per meter, we can calculate the remaining intensity after passing through x meters as follows:
Remaining intensity = Initial intensity * (1 - 0.019)^x
= 1700 * (1 - 0.019)^x
So, the function f(x) to represent the intensity of light when it has passed through x meters of the cable is:
f(x) = 1700 * (1 - 0.019)^x
Part C: In this situation, the scientists want the intensity of light to decrease by 2 lumens per unit length of the cable. Let's analyze if this situation can be represented by a linear function.
If the decrease is constant at 2 lumens per meter, then it can be represented by a linear function. However, in the given question, there is no information about the initial intensity or the rate of decrease. Therefore, we cannot determine if the situation can be represented by a linear function.
However, if we assume the initial intensity to be 1700 lumens, we can write the function to represent this situation. The function f(x) to represent the intensity of light when it has passed through x meters of the cable would be:
f(x) = 1700 - 2x
Where x is the length of the cable in meters. But please note that this assumption might not be accurate without additional information.