Question

Guna has fixed a single door of width 3 feet in his room where as Nathan has fixed a double door, each of width ${\displaystyle 1\frac{1}{2}}$ feet in his room. From the closed position, if each of the single and double doors can open up to 120°, whose door takes a minimum area?

Answer 1

(a) Width of the door that Guna fixed = 3 feet.

When the door is open the radius of the sector = 3 feet

Angle covered = 120°

∴ Area required to open the door

= ${\displaystyle \frac{{120}^{\circ }}{{360}^{\circ }}\times \pi {\text{r}}^2}$

= ${\displaystyle \frac{{120}^{\circ }}{{360}^{\circ }}\times \pi \times 3\times 3}$

= 3π feet²  

(b) Width of the double doors that Nathan fixed = ${\displaystyle 1\frac{1}{2}}$ feet.

Angle described to open = 120°

Area required to open = 2 × Area of the sector

= ${\displaystyle 2\times \frac{{120}^{\circ }}{{360}^{\circ }}\times \pi \times \frac{3}{2}\times \frac{3}{2} {\text{feet}}^2}$

= ${\displaystyle \frac{3\pi }{2}{\text{feet}}^2}$

= ${\displaystyle \frac{1}{2}\left(3\pi \right) {\text{feet}}^2}$

∴ The double door requires the minimum area.