# The figure shows a sector with radius r and angle θ in radians. The total perimeter of the sector is 86 cm.?

Precal Help

The figure is a small right sector of a circle. At the area where the center would be is labeled "Theta" and the bottom of the sector is labeled "r" for radius

(a) Express θ as a function of r.

theta=

b) Express the area of the sector as a function of r.

area = ? cm2

c)For what radius r is the area a maximum? (As usual, decimal approximations will be marked incorrect)

r = cm

d)What is the maximum area? (Give a symbolic answer; decimal approximations will be marked incorrect)

area = cm2

### 1 Answer

- GeronimoLv 77 years agoFavorite Answer
θ = sector angle {radians}

Ps = sector perimeter = 86 cm

As = sector area {cm²}

R = radius {cm}

(a)

Ps = (R • θ) + 2R ... from ——> θ ⁄ (2π) = (arc length) ⁄ (2πR)

arc length = R • θ

86 = (R • θ) + 2R

θ = 2 • (43 – R) ⁄ R ... equation 1

(b)

As = R² • θ ⁄ 2 ... from ——> θ ⁄ (2π) = (As) ⁄ (πR²)

As = R² • (43 – R) ⁄ R ... substituted for θ using eq.1

As = R • (43 – R)

As = - R² + 43R ... which is of the form: y = ax² + bx + c

... where: a = -1 , b = 43 , c = 0

(c)

The vertex is at: R = - b ⁄ (2a) = 43 ⁄ 2 = 21.5 cm ... radius at the vertex

... where As is maximized.

(d)

... and that maximum As value is:

As = - R² + 43R

As = - (21.5²) + 43(21.5)

As = 462.25 cm² ... maximum sector area at: R = 21.5 cm

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