Rate of change

V = (4/3)πr³

dV/dt = (dV/dr)(dr/dt) = 4πr²(dr/dt) ...... (1)

Given dV/dt ∝ A

dV/dt = kA = k(4πr²) ........ (2)

Combine (1) and (2)

4πr²(dr/dt) = k(4πr²)

dr/dt = k



2014-02-10 19:02:43 補充：
b) If the balloon expands at a constant rate of 1 L sec^-1 and if the radius of the balloon is 0.5m, find the rate of increase of
i) its radius
ii) its surface area

from (1) dV/dt = 4πr²(dr/dt)

with dV/dt = 1L/s

1 = 4πr²(dr/dt)

(dr/dt) = 1/(4πr²) = 1/[4π(0.5)²] = 1/π cm/s

A = 4πr²

2014-02-10 19:08:51 補充：
Sorry ! wrong unit!

from (1) dV/dt = 4πr²(dr/dt)

with dV/dt = 1L/s = 10000 cm³/s

1000 = 4πr²(dr/dt)

(dr/dt) = 1000/(4πr²) = 1000/[4π(50)²] = 1/10π cm/s

2014-02-10 19:11:17 補充：
A = 4πr²

dA/dt = (dA/dr)(dr/dt) = 8πr(dr/dt)

dA/dt = 8π(50)(1/10π) = 40 cm² / s