The count of bacteria in a culture was 800 after 10 minutes and 1800 after 40 minutes. What was the initial size of the culture?

The count of bacteria in a culture was 800 after 10 minutes and 1800 after 40 minutes. What was the initial size of the culture?

$$\boxed{b = b_0 * a^{t} } \\\\ b = bacteria \\ b_0=bacteria_{Initial size} \\ t= time\\ \begin{array}{lrcl} \hline (1) & 800 & = & b_0 * a^{10} \quad | \quad t = 10 \quad b = 800\\ (2)& 1800 & = & b_0 * a^{40} \quad | \quad t = 40 \quad b = 1800 \\ \hline \end{array}$$

$$(2):(1) \quad \frac{1800}{800} = \frac{b_0*a^{40} } {b_0*a^{10} }\\\\ \frac{18}{8} = \frac{a^{40} } {a^{10} } = a^{40-10}=a^{30} \quad| \quad \sqrt[30]{}\\\\ \boxed{a=\sqrt[30]{ \frac{18}{8} }} \\\\ (1) \quad b_0 = \dfrac{800}{a^{10}} = \dfrac{800} { \left(\sqrt[30]{ \frac{18}{8} }\right)^{10} } } = \dfrac{800} { \left( \frac{18}{8} }\right)^{\frac{10}{30}} } } = \dfrac{800} { \left( \frac{18}{8} }\right)^{\frac{1}{3}} } } = \dfrac{800} { \frac{ \sqrt[3]{18} } { \sqrt[3]{8} } } = \dfrac{800} { \frac{ \sqrt[3]{18} } { \sqrt[3]{2^3} } } \\\\ = \dfrac{800} { \frac{ \sqrt[3]{18} } { 2 } } = \dfrac{1600}{ \sqrt[3]{18} } = 610.514262695$$

the initial size of the culture is 610.514262695

Asumiing we have a function

N = ab^t    where N is the bacteria count at some time t........note.....when t  = 10, N = 800....and when t = 40, N = 1800...so we have

800 = ab¹⁰     and solving for "a," we have     800/b¹⁰ = a

And we also have

1800 = ab⁴⁰      and substituting, 800/b¹⁰  for a, we have

1800 = (800/b¹⁰)b⁴⁰ →   1800 = 800b³⁰

And dividing both sides by 800  we have

1800/800 = b³⁰  →  9/4 =  b³⁰

And taking the (positive) 30th root of each side, we have that b ≈ 1.0274

And a  = 800/(1.0274)¹⁰  ≈ 611    And this was the initial culture size  (when t = 0, b⁰ = 1)