To find the largest area of the rectangle with its base on the x-axis and its upper vertices on the parabola described by the equation y = 12 – x², we can follow these steps:
1. **Understand the Geometry**: The rectangle’s base will lie along the x-axis, spanning from -a to a for some positive a. Its height will be given by the value of the parabola at x = a, which is y = 12 – a².
2. **Area of the Rectangle**: The area A of the rectangle can be expressed as:
A = base × height
Substituting the values derived:
A = (2a) × (12 – a²) = 24a – 2a³
3. **Maximize the Area**: To find the maximum area, we need to take the derivative of the area function A with respect to a and set it to zero:
A’ = 24 – 6a²
Setting A’ to zero gives:
24 – 6a² = 0
This leads to:
6a² = 24a² = 4a = 2
4. **Calculate Maximum Area**: Now substitute a = 2 back into the area formula:
A = 24(2) – 2(2)³ = 48 – 16 = 32
5. **Conclusion**: Therefore, the largest area of the rectangle that can fit under the parabola y = 12 – x² with its base on the x-axis is 32 square units.