To find the dimensions of the rectangle with the greatest possible area that can be inscribed under the parabola defined by the equation y = 12 – x², with its base on the x-axis, we can follow these steps:
1. Understand the Geometry:
We need to inscribe a rectangle in such a way that the upper corners touch the parabola while the base rests on the x-axis. If we let ‘x’ be half the width of the rectangle, the coordinates of the upper corners of the rectangle will be (-x, y) and (x, y), where ‘y’ is the height of the rectangle. According to the parabola’s equation, the height at any x is given by:
y = 12 – x²
2. Define the Area of the Rectangle:
The area ‘A’ of the rectangle can be expressed as:
A = width × height = (2x) × y = 2x(12 – x²)
So the area function becomes:
A(x) = 2x(12 – x²) = 24x – 2x³
3. Optimize the Area:
To find the maximum area, we need to take the derivative of the area function and set it to zero:
A'(x) = 24 – 6x²
To find the critical points, set the derivative equal to zero:
24 – 6x² = 0
6x² = 24
x² = 4
x = 2 or x = -2 (we only consider x = 2 because we are dealing with the positive width).
4. Calculate Maximum Area:
Next, we calculate the height of the rectangle when ‘x’ is 2:
y = 12 – x² = 12 – 2² = 12 – 4 = 8
5. Determine the Dimensions:
The rectangle dimensions can now be defined:
- Width: 2 * x = 2 * 2 = 4
- Height: y = 8
Final Result:
The dimensions of the rectangle with the greatest possible area that can be inscribed under the parabola are:
- Width: 4 units
- Height: 8 units
This gives us a maximum area of:
Area = Width × Height = 4 × 8 = 32 square units.