To determine which equation best describes the relationship between $x$ and $t$, let's analyze the data. Given that $t$ values are increasing from 1 to 5 and corresponding $x$ values grow very quickly (from 9.38 to 2539.41), this suggests a nonlinear relationship, possibly an exponential or power relationship.

Here's why $x = Bt^q, \, q \geq 2$ is the correct choice:

1. Nonlinear Growth: The values of $x$ increase much more rapidly as $t$ increases. This behavior is consistent with a power function $x = Bt^q$, where $q$ is greater than or equal to 2, indicating faster-than-linear growth.

2. Exponential Growth Consideration: While an exponential function, such as $x = Ae^{kt}$, also exhibits rapid growth, it would typically produce an even more explosive growth pattern with a constant rate of increase, not matching the specific gradual acceleration in $x$ observed here.

3. Power Function with $q \geq 2$: A power function $x = Bt^q$ with $q \geq 2$ fits well because each increase in $t$ appears to result in a polynomial-like growth in $x$, which aligns well with the structure of the data.

4. Other Options:

   • The linear equation $x = mt + b$ would imply a constant rate of increase in $x$ for each unit increase in $t$, which does not match the rapid growth seen here.
   • The equation $x = C t^p$ with $p < 0$ would suggest that $x$ decreases as $t$ increases, which contradicts the data.

Thus, the equation $x = Bt^q$ with $q \geq 2$ best represents the observed data growth pattern.

Would you like more details or have any specific questions?

Here are some further questions you might find interesting:

1. How can we calculate the values of $B$ and $q$ from the data?
2. What types of real-world situations might be modeled by a power function with $q \geq 2$?
3. How would the relationship change if we considered an exponential function?
4. Can we visualize this data and see the curve to confirm the best-fitting equation?
5. How does this power function compare with quadratic or cubic polynomials?

Tip: When determining the best fit for data, consider how quickly the dependent variable grows. Exponential growth often outpaces polynomial growth, but high-degree polynomials can also model rapid changes.