Answer: Jane started growing Sphere 3 hours ago
Step-by-step explanation:
Farm Rod starting population (Rsp) = 2
Farm Sphere starting population (Ssp) = 8
Let´s name "Rh" the quantity of hours since Rod started growing, and
"Sh" the quantity of hours since Sphere started growing.
And, let´s name "R" the population of farm Rod at 8 p.m. and "S" the population of farm Sphere at 8 p.m.
Population of Rod doubles every hour, therefore:
R = [tex]Rsp * 2^{Rh}[/tex]
R = [tex]2(2^{Rh})[/tex]
Population of Sphere is quadrupled every hour, therefore:
S = [tex]Ssp * 4^{Rh}[/tex]
S = [tex]8(4^{Rh})[/tex]
At 8 p.m. Jane found that R = S
Therefore, at 8 p.m:
[tex]2(2^{Rh})[/tex] = [tex]8(4^{Sh})[/tex]
dividing both sides by 2
[tex]2^{Rh} =4(4^{Sh})[/tex]
adding exponents
[tex]2^{Rh} = 4^{Sh+1}[/tex]
[tex]2^{Rh} =2^{2^{Sh+1} }[/tex]
the bases are the same; exponents must be the same
Rh = 2Sh + 2 (equation 1)
And we also know that Jane started growing Rod five hours before Sphere:
Rh = Sh + 5 (equation 2)
Replacing equation 2 into equation 1:
(Sh + 5) = 2Sh + 2
5 - 2 = 2Sh - Sh
3 = Sh, or
Sh = 3
Jane started growing Sphere 3 hours ago.
