Question

A rectangle of te largest possible area is cut out from a semicircle of perimeter 72 cm . Whta is the area of the rectangle

Answer 1

Answer:

525.8 cm²

Step-by-step explanation:

A rectangle of the largest possible area is cut out from a semicircle of perimeter 72 cm . What is the area of the rectangle

Perimeter of Semicircle = 72 cm

Perimeter of semicircle = πr  = 72

=> r = 72/π = 22.93 cm

Let say Height of Rectangle Cut = x

then Length of rectangle cut = 2 *√ (r² - x²)

Area of Rectangle = x * 2 *√ (r² - x²)

Area = A = 2x√ (r² - x²)

dA/Dx =  0

Find x & find Area

dA/dx = 2√ (r² - x²)  + (2x(1/2)/√ (r² - x²))*(-2x)

2√ (r² - x²)  - 2x²/√ (r² - x²) = 0

=> 2(r² - x²)  - 2x² = 0

=> 2r² - 2x² - 2x² = 0

=> 4x² =2r²

=> x = r/√2

Area of reactangle =  x * 2 *√ (r² - x²)

= ( r/√2) * 2 √ (r² - r²/2)

=  ( r/√2) * 2 * r/√2

= r²

= 22.93² = 525.8 cm²