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Choice D is correct. Since the number of bacteria doubles every day, the relationship between $t$ and $y$ can be represented by an exponential equation of the form $y=a{\left(b\right)}^t$, where $a$ is the number of bacteria at the start of the observation and the number of bacteria increases by a factor of $b$ every day. It’s given that there are $\mathrm{44,000}$ bacteria at the start of the observation. Therefore, $a=\mathrm{44,000}$. It’s also given that the number of bacteria doubles, or increases by a factor of $2$, every day. Therefore, $b=2$. Substituting $\mathrm{44,000}$ for $a$ and $2$ for $b$ in the equation $y=a{\left(b\right)}^t$ yields $y=\mathrm{44,000}{\left(2\right)}^t$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This equation represents a situation where the number of bacteria is decreasing by half, not doubling, every day.