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座標平面上の曲線＝x^2（x+3）をCとする。tを正の実数とし、C上の点
P（t，t^2（t＋3））を通り、y軸に平行な直線をlとすると、直線lは曲線CとP以外の2点Q,Rで変わっているとする。このとき、Q,Rを通り、y軸に平行な直線とx軸との交点をそれぞれQ'、R'とし、長方形QRR'Q'の面積をS（t）とす
る。
（1） tのとりうる値の範囲を求めよ。
（2） S（t）をの式で表せ。
（3）S（t）を最大にするtの値とS（t）の最大値を求めよ

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

Explore the solution to a mathematical problem involving the curve C defined by the equation y = x^2(x + 3), points on the curve, and the calculation of a rectangle's area. Understand the range of values for parameter t, express S(t) as a function, and find the maximum value of S(t).

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