Question

Let \(A\) be the region enclosed by the parabola \(y^2 = 2x\) and the line \(x = 24\). Then the maximum area of the rectangle inscribed in the region \(A\) is ________.

Answer 1

Correct Answer: 128

The shaded region in the figure is bounded by the curve and the line \(x = 24\). The coordinates of the points defining this region are \(\left(\frac{b^{2}}{2}, b\right)\) and \((24, b)\).

Step 1: Area Expression
The total area of the shaded region is calculated as:
\[ A = 2 \left( 24 - \frac{b^{2}}{2} \right) b \]

Step 2: Differentiation and Critical Point Calculation
Differentiating the area with respect to \(b\):
\[ \frac{dA}{db} = 2 \left( 24 - \frac{b^{2}}{2} \right) - 2b \cdot \frac{b}{2} \] Setting \(\frac{dA}{db} = 0\) and simplifying to find \(b\):
\[ 48 - 2b^{2} = 0 \quad \Rightarrow \quad b = 4 \]

Step 3: Area Calculation with \(b = 4\)
Substituting \(b = 4\) into the area expression:
\[ A = 2 \left( 24 - \frac{4^{2}}{2} \right) (4) \] \[ A = 2 (24 - 8)(4) \] \[ A = 128 \]

Final Answer:
\[ A = 128 \]