The population of viruses in an influenza culture after t hours is given by the function
x(t) = 4et/2.5
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(t) = y(x(t)) = ln(2*4e^(t/2.5))
= ln(8) + ln(e^(t/2.5))
= ln8 + t/2.5
≈ 0.4t + 2.08
To create the composite function, we need to substitute the expression for x(t) into the function y(x).
Given:
x(t) = 4e^(t/2.5)
y(x) = ln(2x)
Substituting x(t) into y(x):
y(x) = ln(2 * 4e^(t/2.5))
Simplifying:
y(x) = ln(8e^(t/2.5))
Using the logarithmic property ln(ab) = ln(a) + ln(b):
y(x) = ln(8) + ln(e^(t/2.5))
Since ln(e) = 1, we can simplify further:
y(x) = ln(8) + (t/2.5)
Therefore, the composite function that calculates the cost y in dollars for counting the number of viruses in an influential culture after t hours would be:
y(t) = ln(8) + (t/2.5)