Question

A population of honeybees increased at a rate ofbees per week, where the graph of r is as shown. Use Simpson’s Rule with six subintervals to estimate the increase in the bee population during the first 24 weeks.

Short answer

Estimate the increase in the bee population during the first 24 weeks is approximately insects.

Step-by-step solution

Step 1: Applied the population interval (0,24)

r (t) on the interval [0,24]

\(P = \int\limits_0^2 {4r(t)dt} \)

The length is 4.

Step 2: Use the estimate the value of  from graph

From graphvalues

\(\begin{aligned}{l}r(0) = 0\\r(4) = 300\\r(8) = 3,000\\r(12) = 11,000\\r(16) = 4,000\\r(20) = 400\\r(24) = 0\end{aligned}\)

solution of approximate integral

Use the Simpson’s Rule

\({S_n} = \frac{{\Delta t}}{3}\left( {r\left( {{t_0}} \right) + 4r\left( {{t_1}} \right) + 2r\left( {{t_2}} \right) + \cdots + r\left( {{t_n}} \right)} \right)\)

\(\begin{aligned}{l}{S_6} = \frac{{\Delta t}}{3}(r(0) + 4r(4) + 2r(8) + 4r(12) + 2r(16) + 4r(20) + r(24))\\ = \frac{4}{3}(0 + 4 \cdot 300 + 2 \cdot 3,000 + 4 \cdot 11,000 + 2 \cdot 4,000 + 4 \cdot 400 + 0)\\ \approx 81,067in\sec ts\end{aligned}\)