To determine which function best approximates $y = N(e^{3t})$ at $t = 0.2$, we need to evaluate $N(e^{3t})$ at $t = 0.2$ and then compare this value to the outputs of each given function at the same point.

1. Calculate $y = N(e^{3t})$ at $t = 0.2$: $y = N(e^{3 \times 0.2}) = N(e^{0.6}) \approx N(1.8221)$

2. Evaluate each function at $t = 0.2$:

   • (a) $y = N\left(\frac{9}{2}(0.2)^2 + 3(0.2)\right)$ $= N\left(\frac{9}{2}(0.04) + 0.6\right) = N\left(0.18 + 0.6\right) = N(0.78)$

   • (b) $y = N\left(\frac{9}{2}(0.2)^2 + 3(0.2) + 1\right)$ $= N\left(0.18 + 0.6 + 1\right) = N(1.78)$

   • (c) $y = N(3(0.2) + 1)$ $= N(0.6 + 1) = N(1.6)$

   • (d) $y = N\left(\frac{9}{2}(0.2 - 0.2)^2 + 3(0.2 - 0.2)\right)$ $= N(0 + 0) = N(0)$

3. Compare the values:

   • $N(e^{0.6}) \approx N(1.8221)$
   • (a) $N(0.78)$
   • (b) $N(1.78)$
   • (c) $N(1.6)$
   • (d) $N(0)$

The closest value to $N(1.8221)$ is $N(1.78)$ from option (b).

Thus, the function that best approximates $y = N(e^{3t})$ at $t = 0.2$ is (b) $y = N\left(\frac{9}{2}t^2 + 3t + 1\right)$.

Would you like further details or have any questions?

Here are some related questions to consider:

1. How does the value of $N$ affect the approximation?
2. What are other methods to approximate functions at specific points?
3. How would these approximations change if we used a different point $t$?
4. What is the significance of Taylor series in approximating functions?
5. Can you provide a graph comparing these functions near $t = 0.2$?

Tip: When approximating functions, especially exponential ones, Taylor series expansions around a point can provide a systematic way to find suitable polynomial approximations.