An insect population P in a certain area fluctuates with the seasons. It is estimated that P= 17,000 + 4500sin (pi)(t)/52, where t is given in weeks. Determine the number of weeks it would take for the population to initially reach 20,000.

How do I solve this??

Thanks.

Set the function given for P equal to 20,000, and solve for t.

Would the answer be 12 weeks?

thanks.

To solve for the number of weeks it would take for the population initially to reach 20,000, we need to set up and solve the equation:

P = 20,000

First, let's substitute the given formula for P into the equation:

17,000 + 4500sin((πt)/52) = 20,000

To isolate the sine function, we need to move the constant term (17,000) to the other side of the equation:

4500sin((πt)/52) = 20,000 - 17,000

Simplifying:

4500sin((πt)/52) = 3,000

Next, divide both sides by 4500:

sin((πt)/52) = 3,000/4500

Simplifying further:

sin((πt)/52) = 2/3

Now, to solve for t, we need to apply the inverse sine function (sin^(-1)) to both sides of the equation:

sin^(-1)(sin((πt)/52)) = sin^(-1)(2/3)

The inverse sine function undoes the sine function, so we are left with:

(πt)/52 = sin^(-1)(2/3)

Now, multiply both sides by 52/π to isolate t:

t = (52/π) * sin^(-1)(2/3)

Using calculator functions, evaluate the right side to find the value of t. The result will give you the number of weeks it would take for the population initially to reach 20,000.