Question

As a glacier melts, the volume V of the ice, measured in cubic kilometers, decreases at a rate modeled by the differential equation dVdt=kV , where t is measured in years. The volume of the glacier is 400km3 at time t=0 . At the moment when the volume of the glacier is 300km3 , the volume is decreasing at the rate of 15km3 per year. What is the volume V in terms of time

Answer 1

Answer:

the volume in terms of time t ⇒ $\\V = e^{0.05t} + 399$

Explanation:

Given that :

$\frac{dv}{dt}= kv\\\\\int\limits \, \frac{dv}{v}= \int\limits \, kdt\\In v = kt + C_1\\v = e^{kt} + C\\400 = e^{k*0} + C\\400 = 1 + C\\C = 400 -1\\C = 399$

$V = e^{kt} + 399$

When v = 300 ;  $\frac{dv}{dt}= - 15$

then

$\frac{dv}{dt}= kv\\\\-15 = 300*k\\\\k = \frac{-15}{300}\\\\\\k = \frac{-1}{20}\\\\k = -0.05$

∴ $\\V = e^{0.05t} + 399$

Therefore, the volume in terms of time t ⇒ $\\V = e^{0.05t} + 399$