Answer:
a)
[tex]y=400(2.5)^{x}[/tex]
b)
3,814,698
c)
16.08 weeks
Step-by-step explanation:
a)
The question presented here is similar to a compound interest problem. We are informed that there are 400 rice weevils at the beginning of the study. In a compound interest problem this value would be our Principal.
P = 400
Moreover, the population is expected to grow at a rate of 150% every week. This is equivalent to a rate of interest in a compound interest problem.
r = 150% = 1.5
The compound interest formula is given as;
[tex]A=P(1+r)^{n}[/tex]
We let y be the weevil population in any given week x. The formula that can be used to predict the weevil population is thus;
[tex]y=400(1+1.5)^{x}\\\\y=400(2.5)^{x}[/tex]
b)
The weevil population 10 weeks after the beginning of the study is simply the value of y when x = 10. We substitute x with 10 in the equation obtained from a) above;
[tex]y=400(2.5)^{10}\\\\y=3814697.3[/tex]
Therefore, the weevil population 10 weeks after the beginning of the study is approximately 3,814,698
c)
We are simply required to determine the value of x when y is
1,000,000,000
Substitute y with 1,000,000,000 in the equation obtained in a) above and solve for x;
[tex]1000000000=400(2.5)^{x}\\\\2.5^{x}=2500000\\\\xln(2.5)=ln(2500000)\\\\x=\frac{ln(2500000}{ln(2.5)}=16.0776[/tex]
