Answer:
P=7.75t+13
t = 19 weeks
Step-by-step 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:
[tex]\displaystyle y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)[/tex]
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:
[tex]\displaystyle P-P_1=\frac{P_2-P_1}{t_2-t_1}(t-t_1)[/tex]
[tex]\displaystyle P-13=\frac{75-13}{8-0}(t-0)[/tex]
[tex]\displaystyle P-13=\frac{62}{8}t[/tex]
[tex]\displaystyle P-13=7.75t[/tex]
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
