Question

A population of beetles is growing according to a linear growth model. The initial population (week 0) is

Po = 13, and the population after 8 weeks is Ps = 75.
I
Find an explicit formula for the beetle population after a weeks. P. =
After how many weeks will the beetle population reach 161? 20

Answer 1

P=7.75t+13

t = 19 weeks

Explanation:

Linear Modeling

Some situations can be modeled as linear functions. If we are in a situation where a linear model is suitable, then we need two sample points to make the model and predict future behaviors.

The linear function can be expressed in the slope-intercept format:

y = mx + b

Another equation of the line can be used when two points are given.

The equation of a line passing through points (x1,y1) and (x2,y2) can be written as follows:

`\displaystyle y-y_1=(y_2-y_1)/(x_2-x_1)(x-x_1)`

The population of beetles is a situation where we must apply linear modeling. Two points are given. For time t=0, the population is P=13. The point is (0,13). For time t=8, P=75. The point is (8,75).

Find the equation of the line:

`\displaystyle P-P_1=(P_2-P_1)/(t_2-t_1)(t-t_1)`

`\displaystyle P-13=(75-13)/(8-0)(t-0)`

`\displaystyle P-13=(62)/(8)t`

`\displaystyle P-13=7.75t`

The explicit formula is:

P = 7.75t + 13

Now we find when the beetle population is 161:

161 = 7.75t + 13

161 - 13 = 7.75t

7.75t = 148

t = 148/7.75

t = 19 weeks