数学应用题一个球体状的气球被每秒钟13cm^3的速度充满当气球有3800cm^3的空气时直径的瞬时增长速率是多是英语题原题是asphericalballoonisbeingfilledatarateof13cm^3persecond.howfastisthedia

Question

数学应用题 一个球体状的气球被每秒钟13cm^3的速度充满 当气球有3800cm^3的空气时直径的瞬时增长速率是多
是英语题 原题是a spherical balloon is being filled at a rate of 13 cm^3 per second. how fast is the diameter growing at the instant when the balloon has 3800 cm^3 of air?
谢谢了 我特别着急

Answer 1

The volume of the balloon V(t)=13t
Let D(t) be the diameter of the balloon at time t, then V(t) = 4π[D(t)/2]^3 / 3
which gives us D(t) = ∛[(78/π) * t]
Take its derivative, D'(t) = ∛[(26/9π) * t^(-2)]. This is the rate of change of the diameter at time t.
When V(t) = 3800, t = 3800/13
D'(t) = ∛[(26/9π) * (3800/13)^(-2)].
Then just use a calculator and get D'(t) = 0.022