東東 asked in 科學數學 · 1 decade ago

[微分方程]ode微分方程的應用

Assume that a sphere of ice melts at a rate proportional to its surface area,retaining a spherical shape. Interpret melts as a reduction of volume with

respect to time. Determine an expression for the volume of the ice at any time t.

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  • 1 decade ago
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    設冰球半徑為R(t), 體積V(t), 表面積A(t), 初始半徑R(0)=Ro,則

    V(t) = 4π/3 R^3, A(t)=4πR^2,

    由題意知: dV/dt = -kA (k>0為比例常數)

    => 4πR^2 dR/dt = - k * 4πR^2 => dR/dt= -k

    => R=Ro exp(-kt)

    => V= 4π/3 R^3 = 4π/3 Ro^3 exp(-3kt)

    2009-02-20 16:06:10 補充:

    Sorry!我算錯,算成 dR/dt = -kR了!更正:

    dR/dt = -k => R= -kt+ c, R(0)=Ro => R= Ro- kt

    =>V= 4π/3 *R^3 = 4π/3 *(Ro- kt)^3

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