1. ## Linear ODE's

A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\displaystyle \frac{1}{2}$ cm has radius 0.4 cm after 6 months, how long will it take:

(a) For the radius to be $\displaystyle \frac{1}{4}$ cm?

(b) For the volume of the mothball to be half of what it was originally?

2. Originally Posted by Aryth
A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\displaystyle \frac{1}{2}$ cm has radius 0.4 cm after 6 months, how long will it take:

(a) For the radius to be $\displaystyle \frac{1}{4}$ cm?

(b) For the volume of the mothball to be half of what it was originally?
Here's a start: $\displaystyle \frac{dV}{dt} = -k \pi r^2 = -k \pi \left(\frac{3V}{4 \pi}\right)^{2/3}$.

3. Originally Posted by mr fantastic
Here's a start: $\displaystyle \frac{dV}{dt} = k \pi r^2 = k \pi \left(\frac{3V}{4 \pi}\right)^{2/3}$.
The title of the thread is : Linear ODE...

... and that could mean that a linear ODE is required. This goal is achieved writing...

$\displaystyle \frac{dV}{dt} = 4\cdot \pi \cdot r^{2} \frac{dr}{dt} = -4\cdot \pi \cdot r^{2} \cdot \alpha \rightarrow \frac{dr}{dt} = - \alpha$ (1)

Very easy!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. Originally Posted by chisigma
The title of the thread is : Linear ODE...

... and that could mean that a linear ODE is required. This goal is achieved writing...

$\displaystyle \frac{dV}{dt} = 4\cdot \pi \cdot r^{2} \frac{dr}{dt} = -4\cdot \pi \cdot r^{2} \cdot \alpha \rightarrow \frac{dr}{dt} = - \alpha$ (1)

Very easy!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
My DE has the form $\displaystyle \frac{dt}{dV} = f(V)$. Last time I checked, such DE's were classified as linear. However, I do agree that your DE $\displaystyle \frac{dr}{dt} = - \alpha$ provides for a much simpler solution.

5. Thanks a lot. The word problems really mess me up.

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# A substance evaporates at a rate proportional to the exposed surface. If a spherical ball of radius 1/2cm has radius 0.4 cm after 6 months, how long will it take for the radius to be 1/4cm

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