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 1,500 lumens of light enter the cable, the intensity of light decreases by 3.4% per meter of cable.
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 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.
Are these right?
A. No - the decrease in is not a constant
The intensity I of x meters of cable is I = 1500 * 0.964^x
C yes ?
A correct
B correct, except that I think you mean 0.966^x
C yes: C = 1500-5x
Part A: No, this situation cannot be represented by a linear function. In a linear function, the decrease in intensity of light would be constant regardless of the length of the cable. However, in this situation, the decrease in intensity is not a constant rate, but rather a percentage decrease per meter of cable.
Part B: To represent the intensity of light when it has passed through x meters of the cable, we can use an exponential decay function since the decrease in intensity is a percentage decrease. The formula for this function can be written as:
f(x) = 1500 * (1 - 0.034)^x
Where f(x) represents the intensity of light in lumens when it has passed through x meters of the cable.
Part C: No, this situation cannot be represented by a linear function either. In a linear function, the decrease in intensity would be constant regardless of the starting intensity. However, in this situation, the decrease in intensity is dependent on the starting intensity.
To represent the intensity of light when it has passed through x meters of the cable, with a decrease of 5 lumens per unit length, we can use a linear function. The formula for this function can be written as:
f(x) = 1500 - 5x
Where f(x) represents the intensity of light in lumens when it has passed through x meters of the cable.