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.
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 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.
A. No - the decrease in intensity is not a constant, but depends on the previous intensity.
The intensity I of x meters of cable is
I = 1500 * 0.964^x
C yes
you can probably write that one on your own.
Part A:
No, this situation cannot be represented by a linear function. In a linear function, the rate of change or slope is constant. However, in this case, the intensity of light decreases by 3.4% per meter of the cable. The decrease in intensity is not a constant amount, but rather a percentage of the current intensity. Therefore, the relationship between intensity and length is not linear.
Part B:
To represent the intensity of light, f(x), when it has passed through x meters of the cable, we can use an exponential function. The general form of an exponential decay function is:
f(x) = a * (1 - r)^x
Where:
- a is the initial intensity of light (1500 lumens in this case)
- r is the rate of decay, expressed as a decimal (3.4% = 0.034 in this case)
- x is the distance traveled in meters
Therefore, the function to represent the intensity of light, f(x), is:
f(x) = 1500 * (1 - 0.034)^x
Part C:
No, this situation cannot be represented by a linear function either. In a linear function, the rate of change or slope remains constant throughout. However, in this case, the intensity of light is decreasing by a fixed amount (5 lumens) per unit length of the cable. This means that the rate of change or slope of the intensity is not constant but depends on the length of the cable. Hence, the relationship between intensity and length is not linear.
To represent this situation, we can use a linear equation in the form:
f(x) = mx + b
Where:
- m is the rate of decrease in lumens per unit length (-5 lumens in this case)
- x is the distance traveled in units of length
Since 1500 lumens of light enter the cable, the appropriate function to represent this situation would be:
f(x) = -5x + 1500