The number of Drosophila fruit flies in a controlled experiment assumes the following growth pattern when the food source is limited:

N(t) = 120
1+11e−0.4t
where t measured in days.

A. How many fruit files were there in the beginning?

B. What is the limiting number of fruit flies?

C. At what time was the population increasing most rapidly?

D. At what rate is the number of fruit flies increasing after 4 days?

A. To find the number of fruit flies in the beginning, we need to evaluate the equation N(t) when t is equal to 0. Plug in t = 0 into the equation:

N(0) = 120 / (1 + 11e^(-0.4*0))

Simplifying:

N(0) = 120 / (1 + 11e^0)

Since e^0 is equal to 1, the equation can be further simplified:

N(0) = 120 / (1 + 11)

N(0) = 120 / 12

N(0) = 10

So, there were 10 fruit flies in the beginning.

B. To find the limiting number of fruit flies, we need to evaluate the equation N(t) as t approaches infinity. As t approaches infinity, the value of the exponential term e^(-0.4t) becomes close to zero, which does not significantly affect the denominator of the equation. So, the limiting number of fruit flies can be found by evaluating the equation as t approaches infinity:

N(infinity) = 120 / (1 + 11e^(-0.4*infinity))

Since e^(-0.4*infinity) approaches zero, the equation can be simplified to:

N(infinity) = 120 / (1 + 11*0)

N(infinity) = 120 / 1

N(infinity) = 120

Therefore, the limiting number of fruit flies is 120.

C. To find the time at which the population is increasing most rapidly, we need to find the maximum value of the derivative of N(t) with respect to t. The derivative of N(t) with respect to t is given by:

dN/dt = 11 * 0.4 * e^(-0.4t) / (1 + 11e^(-0.4t))^2

To find the maximum value, we can set the derivative equal to zero and solve for t:

11 * 0.4 * e^(-0.4t) / (1 + 11e^(-0.4t))^2 = 0

Simplifying and eliminating the common term of 0.4:

11 * e^(-0.4t) / (1 + 11e^(-0.4t))^2 = 0

Since the exponential term e^(-0.4t) cannot be zero, we can conclude that there is no time at which the population is increasing most rapidly.

D. To find the rate at which the number of fruit flies is increasing after 4 days, we can find the derivative of N(t) with respect to t and then evaluate it at t = 4. The derivative of N(t) with respect to t is given by:

dN/dt = 11 * 0.4 * e^(-0.4t) / (1 + 11e^(-0.4t))^2

Evaluate the derivative at t = 4:

dN/dt = 11 * 0.4 * e^(-0.4*4) / (1 + 11e^(-0.4*4))^2

Simplifying:

dN/dt = 11 * 0.4 * e^(-1.6) / (1 + 11e^(-1.6))^2

Calculating this gives you the rate at which the number of fruit flies is increasing after 4 days.