An investment account was opened with an initial value of $ 890 . The value of

Choice C is correct. It's given that $t$ represents the number of years since the account was opened. Therefore, $\frac{t}{10}$ represents the number of $10$-year periods since the account was opened. Since the value of the account doubles during each of these $10$-year periods, the value of the account can be found by multiplying the initial value by $\frac{t}{10}$ factors of $2$. This is equivalent to $2^{\frac{t}{10}}$. It's given that the initial value of the account is $\$890$. Therefore, the value of the account $M\left(t\right)$, in dollars, $t$ years after the account was opened can be represented by $M\left(t\right)=890{\left(2\right)}^{\frac{t}{10}}$.

Choice A is incorrect. This equation represents the value of an account if the value of the account halves, not doubles, every $10$ years.

Choice B is incorrect. This equation represents the value of an account if the value of the account decreases by $90\%$, not doubles, every $2$, not $10$, years.

Choice D is incorrect. This equation represents the value of an account if the value of the account increases by a factor of $10$, not doubles, every $2$, not $10$, years.