Question

a honeybee population starts with 100 bees and increases at a rate of ( n(t) ) bees per week. what does ( 100+int_{0}^{15} n(t) d t ) represent?
a. the average rate at which the bee population increases over the first 15 weeks.
b. the rate at which the bee population increases after 15 weeks have passed.
c. the total bee population after 15 weeks.
d. the total increase in the bee population after 15 weeks.

Explanation:

• By the Net - Change Theorem, if \(n(t)\) is the population of honeybees at time \(t\) (in weeks), and \(n(0)=100\) (the initial population), and \(n^{\prime}(t)\) is the rate of change of the population. Then \(\int_{0}^{15}n^{\prime}(t)dt=n(15)-n(0)\).
• So \(100+\int_{0}^{15}n^{\prime}(t)dt = n(0)+n(15)-n(0)=n(15)\).

Answer:

C. The total bee population after 15 weeks.