What is the central angle of a sector of a circle whose area and perimeter are, respectively, equal to 209 cm2 and 63 cm if it is given that its radius is a non-negative integer (round up to one decimal place)?
Show answer & explanation
Correct answer: C
Explanation
The correct answer is Option 3 i.e. 49.5°.
We know that,
Perimeter of the sector = 2r + l
here, r= radius, and l = length of the arc
Now,
⇒ 63 = 2r + l......(1)
and,
Area of the sector = 1/2 × l × r
⇒ 209 = 1/2 × l × r
⇒ l × r = 209 × 2
⇒ l × r = 19 × 11 × 2
⇒ l × r = 19 × 22
Putting l and r in eq 1
⇒ 63 = 2r + l
⇒ 63 = 2 × 22 + 19
⇒ 63 = 44 + 19
⇒ 63 = 63 [satisfying]
So, r = 22 and l = 19
Now,
Area of the sector = θ/360 πr2
⇒ 209 = θ/360 × 22/7 × 22 × 22
⇒ θ = (209 × 360 × 7)/(22 × 22 × 22)
⇒ θ = 526680/10648
⇒ θ = 49.46 or 49.5°
Thus, the central angle of a sector of a circle is 49.5°