A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Width of rectangle = 20/(4+π) m, Height of rectangle = 10/(4+π) m, Radius of semicircle = 10/(4+π) m
Explanation
This is an optimization problem from Application of Derivatives. To admit maximum light, we need to maximize the area of the window. The window has a rectangular portion with a semicircle on top. We express area as a function of one variable using the perimeter constraint, then find the critical point where the derivative is zero, and verify it's a maximum using the second derivative test.
Solution Steps
Step 1: Let x be the radius of the semicircular opening and h be the height of the rectangular portion. The width of the rectangle equals 2x (diameter of semicircle).
Step 2: Write the perimeter constraint: Perimeter = 2h + 2x + πx = 10 m (two vertical sides + bottom width + semicircular arc). Therefore, h = (10 - 2x - πx)/2 = 5 - x - πx/2
Step 3: Express area as function of x: Area A = Area of rectangle + Area of semicircle = 2x·h + (1/2)πx² = 2x(5 - x - πx/2) + (1/2)πx² = 10x - 2x² - πx² + (1/2)πx² = 10x - x²(2 + π/2)
Step 4: Find critical point: dA/dx = 10 - 2x(2 + π/2) = 10 - x(4 + π). Setting dA/dx = 0: x = 10/(4 + π)
Step 5: Verify maximum using second derivative: d²A/dx² = -(4 + π) < 0, confirming maximum area.
Step 6: Calculate dimensions: Radius x = 10/(4+π) m, Width = 2x = 20/(4+π) m, Height h = 5 - 10/(4+π) - (π/2)·(10/(4+π)) = 10/(4+π) m