Question

A honeybee population starts with 100 bees and increases at a rate of \(\mathrm{n}^{\prime}(\mathrm{t})\) bees per week. What does \(100+\int_{0}^{15} \mathrm{n}^{\prime}(\mathrm{t}) \mathrm{dt}\) represent?

Short answer

Answer: The expression \(100+\int_{0}^{15} \mathrm{n}^{\prime}(\mathrm{t}) \mathrm{dt}\) represents the total honeybee population after 15 weeks, including the initial 100 honeybees.

Step-by-step solution

Identify the given information

The initial population of honeybees is 100, and the growth rate of the population is given by the function n'(t), which tells us how many bees are added per week. We need to find the total honeybee population after 15 weeks.

Understand the integral

The integral \(\int_{0}^{15} \mathrm{n}^{\prime}(\mathrm{t}) \mathrm{dt}\) represents the cumulative growth of the honeybee population over 15 weeks. The integral finds the area under the n'(t) curve from 0 to 15, which is equal to the total number of bees added during that time.

Compute the integral

In order to find the total population of honeybees after 15 weeks, we need to compute the integral and add it to the initial population: \(100+\int_{0}^{15} \mathrm{n}^{\prime}(\mathrm{t}) \mathrm{dt}\).

Interpret the result

The result of the integral represents the total number of bees added over 15 weeks. So, the expression \(100+\int_{0}^{15} \mathrm{n}^{\prime}(\mathrm{t}) \mathrm{dt}\) is the total honeybee population after the 15 weeks period, including the initial 100 honeybees.