N(t) = 120/(1+11e^(−0.4t)
D. At what rate is the number of fruit flies increasing after 4 days?
To find the rate at which the number of fruit flies is increasing after 4 days, we need to take the derivative of the given function with respect to time.
The given function is N(t) = 120 / (1 + 11e^(-0.4t)). To find the derivative, we can use the quotient rule.
The quotient rule states that if we have a function u(x) divided by another function v(x), the derivative of u(x)/v(x) is given by:
(u'(x) * v(x) - u(x) * v'(x)) / (v(x))^2
Let's use this rule to find the derivative of N(t):
N(t) = 120 / (1 + 11e^(-0.4t))
N'(t) = (120' * (1 + 11e^(-0.4t)) - 120 * (1 + 11e^(-0.4t))') / ((1 + 11e^(-0.4t))^2)
To simplify this further, we need to find the derivatives of the individual parts.
First, let's find the derivative of 120:
120' = 0 (since 120 is a constant)
Next, let's find the derivative of (1 + 11e^(-0.4t)):
(1 + 11e^(-0.4t))' = (1)' + (11e^(-0.4t))'
= 0 + 11(-0.4) * (e^(-0.4t)) (using the chain rule)
= -4.4e^(-0.4t)
Now, let's substitute these values in our initial equation for the derivative:
N'(t) = (0 * (1 + 11e^(-0.4t)) - 120 * (-4.4e^(-0.4t))) / ((1 + 11e^(-0.4t))^2)
= (0 + 528e^(-0.4t)) / (1 + 11e^(-0.4t))^2
Finally, let's find the rate at which the number of fruit flies is increasing after 4 days by substituting t = 4 into the derivative:
N'(4) = (0 + 528e^(-0.4 * 4)) / (1 + 11e^(-0.4 * 4))^2
Now, you can calculate this value using a calculator or a math software to get the rate at which the number of fruit flies is increasing after 4 days.