A rectangle is inscribed in a given semicircle with one of its sides along the diameter. If k times the area of the largest rectangle equals the area of the semicircle then the value of k equals

Question

A rectangle is inscribed in a given semicircle with one of its sides along the diameter. If k times the area of the largest rectangle equals the area of the semicircle then the value of k equals (A) (π/2) (B) (π/4) (C) (π/8) (D) (3 π/4)

Tardigrade Admin · Accepted Answer

Area of rectangle =2 r cos θ ⋅ r sin θ =r2 sin 2 θ ⇒ A max .=r2 ∴ k ⋅ r2=(π r2/2) ⇒ k=(π/2)

Q. A rectangle is inscribed in a given semicircle with one of its sides along the diameter. If $k$ times the area of the largest rectangle equals the area of the semicircle then the value of $k$ equals

$\frac{π}{2}$

$\frac{π}{4}$

$\frac{π}{8}$

$\frac{3 π}{4}$

Area of rectangle $= 2 r cos θ \cdot r sin θ$
$= r^{2} sin 2 θ \Rightarrow A_{m a x .} = r^{2}$
$∴ k \cdot r^{2} = \frac{π r ^{2}}{2} \Rightarrow k = \frac{π}{2}$

Q. A rectangle is inscribed in a given semicircle with one of its sides along the diameter. If k times the area of the largest rectangle equals the area of the semicircle then the value of k equals

2π​

4π​

8π​

43π​