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 `1 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

= `(120^circ)/(360^circ) xx pi"r"^2`

= `(120^circ)/(360^circ) xx pi xx 3 xx 3`

= 3π feet²  

(b) Width of the double doors that Nathan fixed = `1 1/2` feet.

Angle described to open = 120°

Area required to open = 2 × Area of the sector

= `2 xx (120^circ)/(360^circ) xx pi xx 3/2 xx 3/2  "feet"^2`

= `(3pi)/2 "feet"^2`

= `1/2 (3pi)  "feet"^2`

∴ The double door requires the minimum area.