Answer: $15$

Explanation:1. Understand the function: The function $C(t) = 240\left(\dfrac{53}{52}\right)^{t-15} + 7$ gives the estimated number of cephalopods, where $t$ is the number of months since a study began.

2. Set up the equation: We want to find the number of months $t$ when the estimated number of cephalopods $C(t)$ is $247$. So, $247 = 240\left(\dfrac{53}{52}\right)^{t-15} + 7$.

3. Solve for $t$: Subtract $7$ from both sides: $247 - 7 = 240\left(\dfrac{53}{52}\right)^{t-15}$ $240 = 240\left(\dfrac{53}{52}\right)^{t-15}$ Divide both sides by $240$: $\dfrac{240}{240} = \left(\dfrac{53}{52}\right)^{t-15}$ $1 = \left(\dfrac{53}{52}\right)^{t-15}$

4. Solve the exponential equation: For any base $b \neq 0, 1, -1$, if $b^y = 1$, then $y=0$. In our case, the base is $\dfrac{53}{52}$ which is not $0, 1,$ or $-1$. Therefore, the exponent must be zero: $t-15 = 0$ Add $15$ to both sides: $t = 15$

5. Conclusion: The number of cephalopods in the area was estimated to be $247$ at $t=15$ months after the study began.