What is the central angle (in degrees, rounded up to one decimal place) of a sector of a circle whose area and perimeter are, respectively, equal to 209 cm² and 63 cm, given that its radius is a non-negative integer?

Question

Accepted Answer

Let the radius of the circle be r (in cm) and the central angle be θ (in degrees).

Area of sector = (θ/360) × π × r² = 209 ...(1)
Perimeter of sector = 2r + (θ/360) × 2πr = 63 ...(2)

Let us solve for integer values of r.

From (2):
2r + (θ/360) × 2πr = 63
=> 2r + (2πrθ)/360 = 63
=> 2r + (πrθ)/180 = 63
=> (πrθ)/180 = 63 - 2r
=> θ = [180 × (63 - 2r)] / (πr)

Substitute θ in (1):
(θ/360) × π × r² = 209
=> θ × π × r² = 209 × 360
=> θ = (209 × 360) / (π × r²)

Set the two expressions for θ equal:
[180 × (63 - 2r)] / (πr) = (209 × 360) / (π × r²)

Multiply both sides by πr:
180 × (63 - 2r) = (209 × 360) / r

Multiply both sides by r:
180r × (63 - 2r) = 209 × 360

180r × 63 - 180r × 2r = 209 × 360
11340r - 360r² = 75240

Bring all terms to one side:
360r² - 11340r + 75240 = 0

Divide by 12 to simplify:
30r² - 945r + 6270 = 0

Now, solve this quadratic equation:
30r² - 945r + 6270 = 0

Use quadratic formula: r = [945 ± sqrt(945² - 4×30×6270)] / (2×30)
Calculate discriminant:
945² = 893025
4×30×6270 = 4×188100 = 752400
Discriminant = 893025 - 752400 = 140625

sqrt(140625) = 375

So,
r = [945 ± 375] / 60

Possible values:
r₁ = (945 + 375)/60 = 1320/60 = 22
r₂ = (945 - 375)/60 = 570/60 = 9.5

Since r must be a non-negative integer, r = 22 cm.

Now, substitute r = 22 into the formula for θ:
θ = (209 × 360) / (π × 22²)
22² = 484
209 × 360 = 75240
π × 484 ≈ 3.1416 × 484 ≈ 1520.53
θ = 75240 / 1520.53 ≈ 49.48°

Rounded up to one decimal place: 49.5°

Thus, the correct answer is 49.5°.

Answer

Answer

Answer