Mites are discovered in a peach orchard. The Department of Agriculture has determined that the population of mites [tex]t[/tex] hours after the orchard has been sprayed is approximated by:

\[ N(t) = 1900 - 3t \ln(0.131) + 5t \]

where [tex]0 < t < 100[/tex].

Step 2 of 2: What is the maximum number of mites in the peach orchard? Round to the nearest whole number.

The maximum number of mites in the peach orchard, we need to find the maximum value of the function N(t) over the interval 0 < t < 100.To do this, we can take the derivative of N(t) with respect to t and set it equal to zero:

N'(t) = -3ln(0.131) + 5 = 0

Solving for t, we get:

t = (3ln(0.131))/5 ≈ 0.469

To confirm that this value corresponds to a maximum, we can take the second derivative of N(t) with respect to t:

N''(t) = -3/(tln(10)) < 0 for 0 < t < 100

We may infer that the function is concave down and that the critical point we discovered corresponds to a maximum because the second derivative is negative for every t in the interval.

Finally, we can substitute t = 0.469 back into N(t) to find the maximum number of mites:

N(0.469) ≈ 1901

Rounding to the nearest whole number, we get the maximum number of mites in the peach orchard as 1901

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