Quadratic Optimization, Projectile Vertices, and Rectangular Enclosures

Quadratic Trajectory Modeling:
- The height of a launched projectile is modeled by the quadratic expression: $-16t^2 + 152 + 40$
- The variable $t$ represents the elapsed time.
Vertex Principle for Extrema:
- In any optimization problem involving a maximum or a minimum, the underlying mathematical curve is a carabiner.
- Determining an extreme value requires calculating the vertex of the function.
Calculating Time and Peak Altitude:
- The time required for the rocket to reach its maximum height is given directly by the horizontal coordinate ( $t$ -value) of the vertex.
- The maximum height itself is obtained by evaluating the original height function at that calculated vertex time.

Geometric Variables and Boundary Parameters:
- The total perimeter constraint is referenced with parameter $S = 200$ , alongside a boundary span all around of $300$ .
- The width dimension is defined as $x$ .
- The length dimension is defined as $y$ .
- The geometry accounts for multiple length segments, specifically two horizontal boundaries: an upper boundary ( $y$ ) and a lower boundary ( $y$ ).
Formulation of Length in Terms of Width:
- Given the boundary limit of $200$ , an explicit expression for $y$ in terms of $200$ and $x$ is constructed.
- Because there are two length sides (top and down), the remaining perimeter after accounting for the width is divided by $2$ : $y = \frac{200 - \dots}{2}$
Resulting Linear Dimensions:
- Solving the boundary constraints yields: $x = 25$ $y = 50$

Total Enclosure Area:
- The total area encompassing the entire enclosure evaluates to: $1250\,\text{feet}^2$
Subdivision into Adjacent Rectangular Corals:
- The fenced enclosure is designed to form three adjacent rectangular corals.
- The area of each individual rectangular coral is calculated by dividing the total enclosed area by $3$ : $\frac{1250}{3}\,\text{feet}^2$

Quadratic Function Formulation for Area:
- The algebraic model representing the enclosed area is expressed as: $-2x^2 + 100x$
- Written out with an explicit constant term, the quadratic function is: $-2x^2 + 100x + 0$
Optimization Objective:
- The primary objective is to maximize the enclosed area function.
Statistical List Configuration:
- Values representing width or area inputs ( $A$ or $x$ ) are entered into list $L_1$ .
- The corresponding evaluated area function values ( $E(A)$ or $y$ ) are entered into list $L_2$ .
- Computations proceed once all input data pairs are fully populated across $L_1$ and $L_2$ .

Statistical Calculation Output:
- Executing the calculation via the STAT feature produces an output value of: $38.9\dots$
Curve Orientation:
- The parabola opens downward ("open down"), establishing that the vertex corresponds mathematically to a local and absolute maximum.
Vertex Coordinates and Inequality Conditions:
- In standard vertex notation $(h, k)$ :
- The parameter $h$ represents the horizontal ( $x$ ) value of the vertex.
- The parameter $k$ represents the vertical ( $y$ ) value of the vertex.
- The horizontal vertex coordinate satisfies the strict condition: $h > 0$

Audience Inquiry on Function Setup and Operations:
- Question: "Why did you gloves the… So you have your with… And you got negative two x squared plus 100 x. You can put it in, like,."
- Response: At that step in the process, the resulting quadratic expression is $-2x^2 + 100x$ . Technically, this can be viewed with a constant value of zero ( $0$ ). Because the objective is to maximize the area, the variables are assigned to statistical lists by mapping $A$ ( $x$ ) into $L_1$ and $E(A)$ ( $y$ ) into $L_2$ .

Subsequent Group Work Activity:
- A group work assignment directly utilizes the quadratic vertex optimization techniques and area partitioning formulas established during the session.