Question

Consider the region \[ R = \left\{ (x, y): x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0 \right\}. \] The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \) is:

• A. \( \frac{625}{111} \)
• B. \( \frac{730}{119} \)
• C. \( \frac{567}{121} \)
• D. \( \frac{821}{123} \)

Hint:

When solving optimization problems involving areas, differentiate the area function with respect to the variable (in this case, \( x \)) and solve for the critical points. After finding the critical points, check whether they correspond to a maximum by examining the second derivative or using other methods.

Answer 1

The Correct Option is C

To find the area of the largest rectangle inscribed within the region \(R = \left\{ (x, y): x \leq y \leq 9 - \frac{11}{3} x^2, x \geq 0 \right\}\), we begin by understanding the boundary conditions that define the region. 

1. First, analyze the boundaries:
   • \(y = x\) is the line \(y = x\).
   • \(y = 9 - \frac{11}{3}x^2\) is a downward-opening parabola intersecting the y-axis at \(y = 9\).
   • \(x \geq 0\) restricts the region to the positive x-axis.
2. To find the intersection points of the lines, set \(y = x\) equal to \(y = 9 - \frac{11}{3}x^2\):
   • \(x = 9 - \frac{11}{3}x^2\).
   • Rearrange: \(\frac{11}{3}x^2 + x - 9 = 0\).
3. Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = \frac{11}{3}\), \(b = 1\), and \(c = -9\):
   • Discriminant \(\Delta = b^2 - 4ac = 1 + \frac{44}{3} \times 9 = 121\).
   • \(x = \frac{-1 \pm \sqrt{121}}{\frac{22}{3}} = \frac{-1 \pm 11}{\frac{22}{3}}\).
   • Thus, possible \(x\) are \(x = \frac{10}{\frac{22}{3}} = \frac{30}{11}\) (taking the positive square root for a valid x-intersection).
4. Now that we have the boundaries of the region, consider the largest rectangle:
   • The height, \( h = x \).
   • The width can be determined by considering \(y = 9 - \frac{11}{3} x^2\):
   • Therefore, at \(x = \frac{30}{11}\), maximum possible \(y = x\) for largest rectangle occurs when intersects y of parabola: \(9 - \frac{11}{3}x^2\).
5. Calculate maximum area of rectangle:
   • Height \(h = \frac{30}{11}\), Width \(w = 9 - \frac{11}{3}\left(\frac{30}{11}\right)^2\).
   • Substitute \(x = \frac{30}{11}\) and compute width: 
     \(w = 9 - \frac{11}{3} \left( \frac{900}{121} \right)\)
   • Simplifying: 
     \(w = 9 - \frac{9900}{363} = \frac{3267 - 9900}{363} = \frac{567}{121}\).

The largest such rectangle has an area of \(\frac{567}{121}\)

Answer 2

Given the curve: 

\[ y = 9t - \frac{11t^3}{3} \] and the area \( A \) of the rectangle inscribed under this curve is: \[ A = t \left( 9t - \frac{11t^3}{3} \right) = 9t^2 - \frac{11t^4}{3}. \]

Step 1: Differentiate the area function

To find the maximum area, we differentiate the area function with respect to \( t \): \[ \frac{dA}{dt} = 18t - \frac{44t^3}{3}. \]

Step 2: Set the derivative equal to zero

Set \( \frac{dA}{dt} = 0 \) to find critical points: \[ 18t - \frac{44t^3}{3} = 0. \] Multiply through by 3 to eliminate the fraction: \[ 54t - 44t^3 = 0. \] Factor out \( t \): \[ t(54 - 44t^2) = 0. \] Thus, \( t = 0 \) or \( t = \pm \frac{9}{11} \).

Step 3: Determine the maximum point

From the graph and further analysis, we determine that the maximum occurs at \( t = \frac{9}{11} \).

Step 4: Calculate the maximum area

Substituting \( t = \frac{9}{11} \) into the area formula: \[ A = \frac{9}{11} \left( 9 - \frac{11 \cdot 9^3}{3 \cdot 11^3} \right). \] Simplifying: \[ A = \frac{9}{11} \left( 9 - \frac{81}{121} \right) \] \[ A = \frac{9}{11} \times \frac{63}{11} = \frac{567}{121}. \]

Final Answer:

The largest area is \( \frac{567}{121} \).