Question

Question 5A population of beetles is growing according to a linear growth model. Initially, there were 32 beetles, and 10 weeks later, there were 38 beetles.(a) Write a linear model to describe the number of beetles over time, using weeks as the unit of time.Pt = (b) How many beetles are there expected to be 13 weeks after the initial point? beetles(c) When do you expect the number of beetles to reach 85? Round your answer to the nearest week.After weeks

Answer 1

(a) A linear model for the population P(t) after t weeks have passed is:

P(t) = at + b

where b is the initial population. In this case, b = 32.

Since for t = 10, P(t) = 38, we have:

38 = a * 10 + 32

38 - 32 = 10a

6 = 10a

6/10 = a

a = 0.6

Therefore, the linear model is:

P(t) = 0.6t + 32

(b) Now, after 13 weeks, we have:

P(13) = 0.6 * 13 + 32 = 7.8 + 32 = 39.8

Therefore, after 13 weeks, approximately 40 beetles are expected.

(c) Now, we need to find t for which P(t) = 85:

85 = 0.6t + 32

85 - 32 = 0.6t

53 = 0.6t

t = 53/0.6

t ≅ 88.333

t ≅ 88

Therefore, rounding the result to the nearest week, the answer is:

After 88 weeks.