Question

suppose ( p(t) ) represents the population of a certain mosquito colony, where ( t ) is measured in days. the current population of the colony is known to be 566 mosquitos; that is, ( p(0)=566 ). if ( p^{prime}(0)=178 ) mosquitos per day, estimate the size of the population in 17 days. mosquitos (round to nearest whole number) question help: video message instructor submit question jump to answer

Explanation:

Step1: Use the linear approximation formula

The linear approximation formula is \(P(t)\approx P(0)+P^{\prime}(0)t\). Here, \(P(0) = 566\), \(P^{\prime}(0)=178\), and \(t = 17\).

Step2: Substitute the values into the formula

Substitute the values into the formula: \(P(17)\approx566+178\times17\).
First, calculate \(178\times17=(180 - 2)\times17=180\times17-2\times17=3060 - 34 = 3026\).
Then, \(P(17)\approx566+3026\).

Step3: Calculate the result

\(P(17)\approx566 + 3026=3592\).

Answer:

3592