During an epidemic, the number of people who have never had the disease and who are not immune (they are susceptible) decreases exponentially according to the function where t is time in days.
f(t)=15,000e^-.05t
Determine how long it will take for the initial number of people susceptible to decrease to half the amount
you want t such that
e^-.05t = 0.5
-.05t = ln(0.5)
t = ln(0.5)/(-.05) = 13.86
To determine how long it will take for the initial number of people susceptible to decrease to half the amount, we need to find the value of t when f(t) is equal to half the initial number.
Given the function f(t) = 15,000e^(-0.05t), let's find half the initial number:
Half the initial number = 15,000 / 2 = 7,500
Now, we will set f(t) equal to 7,500 and solve for t:
7,500 = 15,000e^(-0.05t)
To isolate the exponential term, we divide both sides by 15,000:
7,500 / 15,000 = e^(-0.05t)
Simplifying, we get:
0.5 = e^(-0.05t)
To get rid of the exponential function, we take the natural logarithm (ln) of both sides:
ln(0.5) = ln(e^(-0.05t))
Using the property of logarithms, ln(e^x) = x, we simplify further:
ln(0.5) = -0.05t
Finally, we solve for t by dividing both sides by -0.05:
t = ln(0.5) / -0.05
Using a calculator, we can evaluate this expression to find the value of t, which represents the time it will take for the initial number of people susceptible to decrease to half the amount.