The problem provided appears to deal with a curve and a specific configuration of points and areas on the coordinate plane. Here is a general translation and breakdown of the problem:

Problem (Translated):

Consider the curve $C$ given by the equation $y = x^2(x + 3)$. Let $t$ be a positive real number, and let $P(t, t^2(t + 3))$ be a point on the curve $C$. A vertical line $l$ passes through point $P$ and intersects the curve at two additional points $Q$ and $R$, apart from $P$.

Define the points $Q'$ and $R'$ as the intersections of the vertical lines through $Q$ and $R$ with the x-axis, respectively. Let $S(t)$ represent the area of the rectangle formed by the points $Q, R, R', Q'$.

Tasks:

1. Find the possible range of values for $t$.
2. Express $S(t)$ as a function of $t$.
3. Find the value of $t$ that maximizes $S(t)$ and determine the maximum value of $S(t)$.

Let me go ahead and solve the problem step by step.

Part 1: Finding the range of $t$

Since $t$ is a positive real number, we can conclude that $t \in (0, \infty)$. This is because the problem states that $t$ is a positive real number and doesn't provide any further restrictions.

Part 2: Expressing $S(t)$

The rectangle's area $S(t)$ is based on the distances between the points $Q, R, Q',$ and $R'$. We need to determine these points based on the properties of the curve and the intersections. This will require solving for the points of intersection of the vertical line through $P$ with the curve.

Let me perform the necessary calculations for this part.import sympy as sp

Define variables and functions

t = sp.symbols('t') x = sp.symbols('x') y = x**2 * (x + 3)

Point P(t, y(t))

P_y = y.subs(x, t)

Set up the vertical line through point P at x = t

This line intersects the curve at other points Q and R.

We solve for other points where the curve intersects the vertical line x = t

Define the equation of the curve y = x^2(x + 3)

Solve y = y for intersections

curve_eq = sp.Eq(y, P_y)

Solve for x

solutions = sp.solve(curve_eq, x)

solutionsThe solutions to the intersection points between the curve $y = x^2(x + 3)$ and the vertical line $x = t$ are:

1. $x = t$ (This is the point $P$.)
2. Two additional solutions for $x$ are: $x = -\frac{t}{2} - \frac{\sqrt{3}}{2} \sqrt{-t^2 - 2t + 3} - \frac{3}{2},$ $x = -\frac{t}{2} + \frac{\sqrt{3}}{2} \sqrt{-t^2 - 2t + 3} - \frac{3}{2}.$

These are the $x$-coordinates of the points $Q$ and $R$, where the vertical line intersects the curve.

Next Step

The area of the rectangle $S(t)$ is determined by the horizontal distance between the points $Q$ and $R$ and the height difference between these points along the $y$-axis. I'll now compute the horizontal distance and the height difference to derive the formula for $S(t)$.

Let me continue with the calculation.# Define the two non-trivial solutions (Q and R points) x_Q = solutions[1] x_R = solutions[2]

Calculate the horizontal distance between Q and R

horizontal_distance = sp.simplify(x_R - x_Q)

Now we calculate the y-coordinates of the points Q and R by plugging the x-values into the curve equation

y_Q = y.subs(x, x_Q