Here is a detailed solution to the problems in the uploaded image.

Problem 3: Banker modeling expected value

Formula:

$v = \frac{1}{2}n^3 - 2n^2 + n + 3$

Key Features:

1. Zeros: Find where $v = 0$ by solving $\frac{1}{2}n^3 - 2n^2 + n + 3 = 0$.
2. Extrema: Compute the derivative $v'(n)$, set $v'(n) = 0$, and solve for $n$ to find critical points. Then, determine whether these are maxima or minima using the second derivative test.

Problem 4: Plant height modeling

Formula:

$f(x) = 1.5x^2 - 2x + 1$

Key Features:

1. Zeros: Find where $f(x) = 0$ by solving $1.5x^2 - 2x + 1 = 0$.
2. Extrema: Compute $f'(x)$, solve $f'(x) = 0$, and analyze the critical points to find maxima or minima.

Problem I: Tree height equation

Equation:

$3n + 8 = 27$

1. Solve for $n$: $3n + 8 = 27 \quad \Rightarrow \quad 3n = 19 \quad \Rightarrow \quad n = \frac{19}{3} \approx 6.33 \, \text{years}.$

Interpretation of "3":

• The correct option is B: "The average number of feet the tree grew per year." The constant "3" represents the growth rate per year in feet.

Problem II: Quadratic equation

Equation:

$3a^2 + 14a - 5 = 0$

1. Use the quadratic formula: $a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 3$, $b = 14$, and $c = -5$.

   Substituting: $a = \frac{-14 \pm \sqrt{14^2 - 4(3)(-5)}}{2(3)} = \frac{-14 \pm \sqrt{196 + 60}}{6} = \frac{-14 \pm \sqrt{256}}{6}.$

   Simplify: $a = \frac{-14 \pm 16}{6}.$

   Two solutions: $a = \frac{-14 + 16}{6} = \frac{2}{6} = \frac{1}{3}, \quad a = \frac{-14 - 16}{6} = \frac{-30}{6} = -5.$

   Solutions: $a = \frac{1}{3}, \, a = -5.$

Would you like further clarifications on any part or help graphing the functions?

Expanding Questions:

1. Can you describe the significance of extrema in real-life modeling contexts?
2. Why is it important to determine zeros for polynomial functions like these?
3. How would the derivative of the banker’s formula help us understand growth trends?
4. Could the tree growth problem be modeled using a quadratic equation? Why or why not?
5. What does the discriminant tell us about the nature of solutions to quadratic equations?

Tip:

Always check the units and context of a mathematical model to ensure proper interpretation of constants and variables.