Question

The braconid wasp preys on the azuki bean weevil. This predator-prey relationship causes the wasp population to fluctuate. The

wasp population after t generations is given by

w = 3500 + 2000 sin(πt/3)

After how many generations is the wasp population at a minimum for the first time?

(Give your answer to the nearest whole number.)
1 =
generation(s)

What is the minimum number? (Give an exact answer)

Minimum wasp population :
When is the first time the wasp population reaches 5400?
(Give your answer to the nearest whole number )
T=

Answer 1

The wasp population is at a minimum after 5 generations, with a minimum population of 1500 wasps. The population reaches 5400 wasps for the first time after approximately 3 generations.

Step-by-step explanation:

To determine when the wasp population is at a minimum for the first time, we look for when the sine function in the equation w = 3500 + 2000 sin(πt/3) reaches its minimum value. The sine function reaches its minimum value of -1. Therefore, we need to solve for when sin(πt/3) = -1. Using the unit circle or sine properties, we know that sin(θ) = -1 when θ = 3π/2 (or 270 degrees). So:

πt/3 = 3π/2
t = (3π/2) × (3/π)
t = 9/2 = 4.5
To the nearest whole number, t = 5 generations.

The minimum wasp population occurs when sin(πt/3) = -1, which corresponds to w = 3500 + 2000(-1) = 1500 wasps.

To find when the population reaches 5400, we set w to 5400 and solve for t:

5400 = 3500 + 2000 sin(πt/3)
1900 = 2000 sin(πt/3)
sin(πt/3) = 1900/2000
sin(πt/3) = 0.95

We find the value of t that corresponds to sin(πt/3) = 0.95. Using a calculator to find the inverse sine (arcsin) we get:

πt/3 = arcsin(0.95)
t = 3 × arcsin(0.95) / π

Using a calculator, t ≈ 3.32. To the nearest whole number, T = 3 generations.