1 - exponential growth formula:
[tex]y=a(1+r)^x[/tex]where:
y is the number of Beatles
a is the initial population of beetles
r is the rate (as a decimal)
x is time in moths
Replacing with a = 850, and r = 0.07, we get:
[tex]\begin{gathered} y=850(1+0.07)^x^{} \\ y=850(1.07)^x \end{gathered}[/tex]2 - Replacing with x = 11:
[tex]y=850(1.07)^{11}\approx1789\text{ beetles}[/tex]3 - Replacing with y = 100,000:
[tex]\begin{gathered} 100000=850(1.07)^x \\ \frac{100000}{850}=(1.07)^x \\ \ln (\frac{100000}{850})=x\cdot\ln (1.07) \\ \frac{\ln (\frac{100000}{850})}{\ln (1.07)}=x \\ 70.5\text{ months }\approx\text{ x} \end{gathered}[/tex]12 months is 1 year, then 70.5 months is equivalent to 70.5/12 = 5.875 years
