.0. asked in 科學數學 · 1 decade ago

Apply of Differentiation (急!!)

A piece of wire of length k cm is bent to from a sector. Find the maximum area of the sector.

2 Answers

Rating
  • 1 decade ago
    Favorite Answer

    Suppose that the radius of sector is x cm, then the arc length will be (k - 2x) cm.

    Moreover, the angle subtended at the centre of the sector is:

    θ = (k - 2x)/x = (k/x - 2) radians

    Applying the formula of sector area:

    A = x2θ/2

    = x2(k/x - 2)/2

    = (kx/2 - x2) cm2

    Taking differentiation of A w.r.t. x:

    dA/dx = (k/2 - 2x)

    d2A/dx2 = - 2

    When dA/dx = 0, x = k/4 and since d2A/dx2 < 0, it is for sure that x = k/4 will give a maximum of A.

    Hence the max. area is:

    Amax = [k(k/4)/2 - (k/4)2] = k2/16 cm2

    Source(s): My Maths knowledge
  • wy
    Lv 7
    1 decade ago

    Let radius of the sector = r and angle of sector = x.

    k = rx + 2r = r( x + 2)......... (1) ( x in radian)

    Area, A = r^2x/2.......(2)

    From (1) 0 = ( x + 2) + r dx/dr, so dx/dr = -(x + 2)/r.

    From (2)

    dA/dr = [r^2 dx/dr + 2rx]/2

    = [- r^2(x + 2)/r + 2rx]/2

    = [-(x + 2)r + 2rx]/2

    = (- 2r + rx)/2.

    Put dA/dr = 0, we get

    2r = rx

    x = 2.

    when x = 2, from (1), r = k/4.

    so A max. = (k/4)^2(2/2) = (k/4)^2.

Still have questions? Get your answers by asking now.