Exponential growth of a bacteria population

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A bacteria population grows as $y=b\cdot a^{x}$ ($x$ in hours, $y$ the count). At $x=0$ there are $200$, after $5$ hours $1800$.

• (a) find $b$ and $a$;
• (b) compute the count after $8$ hours;
• (c) after how many hours does it reach $10\,000$?

Solution

(a) $b=200$ (initial value); $a^5=\dfrac{1800}{200}=9\Rightarrow a=\sqrt[5]{9}\approx1{,}552$.

(b) $y(8)=200\cdot 9^{8/5}\approx 6727$.

(c) $10\,000=200\cdot 9^{x/5}\Rightarrow 9^{x/5}=50\Rightarrow x=5\log_9 50\approx 8{,}90$ hours.

$\boxed{b=200,\ a=\sqrt[5]{9}\approx1{,}55,\ y(8)\approx6727,\ x\approx8{,}90\ \text{h}}$