Question

A population of beetles is growing according to a linear growth model. The initial population (week 0) was P0=3, and the population after 8 weeks is P8=67.

a. Find an explicit formula for the beetle population in week n b. After how many weeks will the beetle population reach 187?

Answer 1

a. To find an explicit formula for the beetle population in week n, we need to use the two given data points to determine the slope and y-intercept of the linear growth model.

First, we can find the slope (m) using the formula:

m = (P8 - P0)/(8 - 0) = (67 - 3)/(8 - 0) = 8

Next, we can find the y-intercept (b) by substituting the slope and one of the data points into the point-slope form of a linear equation:

y - P0 = m(x - 0)

y - 3 = 8x

y = 8x + 3

Therefore, the explicit formula for the beetle population in week n is:

Pn = 8n + 3

b. To find out after how many weeks the beetle population will reach 187, we can set Pn = 187 in the formula we found in part (a), and solve for n:

8n + 3 = 187

8n = 184

n = 23

Therefore, the beetle population will reach 187 after 23 weeks.