Finding the Area of the Largest Inscribed Rectangle in a Semi-Circle
To determine the area of the largest rectangle inscribed in a semi-circle with a radius of 5, we can use some geometry and calculus.
1. Understanding the Geometry
Consider a semi-circle with center at the origin (0,0) in the Cartesian plane, extending from -5 to 5 on the x-axis. The equation of the semi-circle is:
y = √(25 - x²)
The rectangle will have its base on the x-axis and its top corners on the semi-circle. Let’s say the rectangle extends from (-x, 0) to (x, 0) and its height is determined by the semi-circle at that x value, which is √(25 – x²).
2. Area of the Rectangle
The area (A) of the rectangle can be expressed as:
A = width × height
Here, the width is 2x (from -x to x), and the height is √(25 – x²). Therefore:
A = 2x * √(25 - x²)
3. Maximizing the Area
To find the value of x that maximizes the area, we can differentiate the area with respect to x:
Let A = 2x * √(25 - x²)
Using the product rule, we differentiate A:
dA/dx = 2√(25 - x²) + 2x * (1/2)(25 - x²)^(-1/2)(-2x) = 2√(25 - x²) - (2x²/(√(25 - x²)))
Setting this derivative equal to zero gives us the critical points:
2√(25 - x²) - (2x²/(√(25 - x²))) = 0
4. Solving the Equation
Rearranging the equation leads to:
2√(25 - x²) = (2x²/(√(25 - x²)))
Squaring both sides to eliminate the square root (and assuming x > 0):
4(25 - x²) = 4x²
Simplifying gives:
100 - 4x² = 4x²
100 = 8x²
x² = 12.5
x = √12.5 = 5/√2 = 5√2/2
5. Calculating the Maximum Area
Now that we have x, we can find the corresponding height:
height = √(25 - (5√2/2)²) = √(25 - 12.5) = √12.5
Then, the area of the rectangle can be calculated as:
A = 2(5√2/2) * √(12.5) = 5√2 * √12.5
Consolidating the square roots:
A = 5√(25) = 5 * 5 = 25
Conclusion
Thus, the area of the largest rectangle that can be inscribed in a semi-circle of radius 5 is 25 square units.