The population of viruses in an influenza culture after t hours is given by the function
x(t) = 3et/1.7
The cost y in dollars for a new automated microscope to count x viruses in a sample is
y(x) = ln 2x
Create the composite function that calculates the cost y in dollars of counting the number of
viruses in an influential culture after t hours.
y(x(t))
= ln(2x(t))
= ln (2*3e^(t/1.7))
= ln6 + ln e^(t/1.7)
= t/1.7 + ln6
To create the composite function that calculates the cost y in dollars of counting the number of viruses in an influenza culture after t hours, we need to substitute the expression for x(t) into the function y(x).
Given:
x(t) = 3e^(t/1.7) (function for the population of viruses after t hours)
y(x) = ln(2x) (function for the cost of counting x viruses)
To substitute x(t) into y(x), we replace each occurrence of x in y(x) with x(t).
y(x(t)) = ln(2x(t))
Now, substitute x(t) = 3e^(t/1.7) into y(x(t)):
y(x(t)) = ln(2 * 3e^(t/1.7))
Simplify the expression:
y(x(t)) = ln(6e^(t/1.7))
Since ln(a * b) = ln(a) + ln(b), we can further simplify the expression:
y(x(t)) = ln(6) + ln(e^(t/1.7))
Since ln(e^x) = x, we can simplify it to:
y(x(t)) = ln(6) + (t/1.7)
Therefore, the composite function that calculates the cost y in dollars of counting the number of viruses in an influenza culture after t hours is:
y(x(t)) = ln(6) + (t/1.7)