To find the number of bacteria after 512 hours using the function \( B(h) = 82(1.25)^h \), we need to calculate \( B(512) \).
\[ B(512) = 82(1.25)^{512} \]
Now, calculating \( (1.25)^{512} \):
Since \( (1.25)^{512} \) will be a very large number, we can use a calculator or computational tool for the precise computation.
Calculating gives:
\[ (1.25)^{512} \approx 1.083 \times 10^{17} \]
Now substituting back into the equation for \( B(512) \):
\[ B(512) \approx 82 \times 1.083 \times 10^{17} \]
Calculating this yields:
\[ B(512) \approx 8.8726 \times 10^{18} \]
This number is astronomically large, far exceeding any of the options given (564, 257, 279, 280).
Given that the output from the function is likely too vast for realistic bacteria growth in a normal context, it might be important to consider if there is a misinterpretation about the parameters or scale of the output.
If we are just looking for the nearest bacteria based on the severely exponential growth observed in the calculation after inputting 512 hours, we still affirm that:
\[ B(512) \text{ is indeed extraordinarily larger than provided options.} \]
However, if we disregard the exponential and consider a more realistic range check or a human error in input hours, please clarify further on whether the exponent is confined or intended differently.
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