Weare told that the growth is linear. Since the initial population is 3 , we have the following point on the plane represented by number of beetler on the vertical axis, and number of weeks in the horizontal axis:
(0, 3) (week zero we had 3 beetles)
SInce after 8 weeks there were 67 beetles, then the next point is:
(8, 67)
Now, with these two points we find the slope of the segment that joins them with the formula:
slope = (y2 - y1) / (x2 - x1)
in our case:
slope = (67 - 3) / (8 - 0)= 64/8 = 8
so the slope of the line is 8, and its y-intercept is 3 (point at which the line crosses the y axis (0,3) as found above:
Then the equation that represents the beetle population is:
Population = 8 x + 3
where x represents the number of weeks. Therefore for the population to grow to eacj 187 beetles, we have to solve for x in the equation:
187 = 8 x + 3
subtract 3 from both sides
187 - 3 = 8 x
184 = 8 x
divide by 8 to isolate x
184 / 8 = x
x = 23
so, it takes a total of 23 weeks, for the population to reach 187 beetles.