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? asked in Education & ReferenceHomework Help · 7 years ago

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

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  • 7 years ago
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       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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