Question

A population of beetles are growing accordingto a linear growth model. The initial population (week 0) isPo = 5, and the population after 7 weeks is P = 82.Find an explicit formula for the beetle population after n weeks..Pn-After how many weeks will the beetle population reach 258?weeks

Answer 1

P(n) = 5 + 11n

n = 23 weeks

Step-by-step explanation:

The equation for the population as a linear growth model has the form

P = P0 + an

Where P0 is the initial population, n is the number of weeks and a is the rate of increase per week. We know that P0 = 5, so

P = 5 + an

Additionally, when n = 7 the value of P = 82, so we can use this to find the value of a as follows

82 = 5 + a(7)

82 = 5 + 7a

82 - 5 = 5 + 7a - 5

77 = 7a

77/7 = 7a/7

11 = a

Therefore, the equation for the population after n weeks is

P(n) = 5 + 11n

Finally, to know the number of weeks to reach a population of 258, we need to replace P by 258 and solve for n, so

258 = 5 + 11n

258 - 5 = 5 + 11n - 5

253 = 11n

253/11 = 11n/11

23 = n

So, after 23 weeks the population will be 258.