To find the number of bacteria after \(5\frac{1}{2}\) hours, we first need to convert \(5\frac{1}{2}\) to a decimal. This is equal to \(5.5\) hours.
Now, we can plug \(h = 5.5\) into the function \(B(h) = 82(1.25)^{h}\):
\[ B(5.5) = 82(1.25)^{5.5} \]
Now, let's calculate \( (1.25)^{5.5} \):
\[ (1.25)^{5.5} \approx (1.25^5)^{1} \times (1.25^{0.5}) \approx 3.0517578125 \times 1.1180339887 \approx 3.41421118 \]
Now calculate \(B(5.5)\):
\[ B(5.5) \approx 82 \times 3.41421118 \approx 280.5926 \]
Rounding to the nearest bacteria, we get:
\[ B(5.5) \approx 281 \]
So, the number of bacteria after \(5 \frac{1}{2}\) hours will be approximately 281 bacteria.
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