suppose ( p(t) ) represents the population of a certain mosquito colony, where ( t ) is measured in days…

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

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

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