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# Writing differential equations, C4. Tweet

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1. Writing differential equations, C4.
I don't recall being taught this, yet it has cropped up in a few past papers.

A giant snowball is melting. The snowball can be modelled as a sphere whose
surface area is decreasing at a constant rate with respect to time. The surface area of
the sphere is A cm2 at time t days after it begins to melt.

(a) Write down a differential equation in terms of the variables A and t and a constant k,
where k > 0 , to model the melting snowball.

How do you know what to do?
2. Re: Writing differential equations, C4.
(Original post by hash007)
The snowball can be modelled as a sphere whose
surface area is decreasing at a constant rate with respect to time.
What should be the rate of change of area with respect to time? What does the decreasing implies here?
3. Re: Writing differential equations, C4.
(Original post by Zishi)
What should be the rate of change of area with respect to time? What does the decreasing implies here?
dA/dT

but the answer is dA/dT = -k

Why is it -k?
4. Re: Writing differential equations, C4.
(Original post by hash007)
dA/dT

but the answer is dA/dT = -k

Why is it -k?
It's decreasing at a proportional rate. If it was increasing at a proportional rate it'd be positive k.
5. Re: Writing differential equations, C4.
(Original post by hash007)
dA/dT

but the answer is dA/dT = -k

Why is it -k?
That's what decreasing implies here. Negative sign has to be put here to show that the rate is decreasing because the question already states that k is greater than 0.
6. Re: Writing differential equations, C4.
You are told the surface area is decreasing at a constant rate wrt time.

If represents the change in surface area wrt time and k>0 then

7. Re: Writing differential equations, C4.
(Original post by Zishi)
That's what decreasing implies here. Negative sign has to be put here to show that the rate is decreasing because the question already states that k is greater than 0.
I see, that's not too hard then. The wording 'k>0' confused me, but it has to be < 0 otherwise the rate of change would be positive, which can't happen as the area is decreasing. Cheers!
8. Re: Writing differential equations, C4.
Actually I don't know what to do next

Initially, the radius of the snowball is 60 cm, and 9 days later, the radius has halved.
Show that A = 1200pi (12 - t).
(You may assume that the surface area of a sphere is given by A = 4pi r^2 , where r is
9. Re: Writing differential equations, C4.
(Original post by hash007)
Actually I don't know what to do next

Initially, the radius of the snowball is 60 cm, and 9 days later, the radius has halved.
Show that A = 1200pi (12 - t).
(You may assume that the surface area of a sphere is given by A = 4pi r^2 , where r is
You've to solve the differential equation by taking integral of both sides(after you take dt on right hand side, of course). What you'll get after that?
10. Re: Writing differential equations, C4.
(Original post by hash007)
Actually I don't know what to do next

Initially, the radius of the snowball is 60 cm, and 9 days later, the radius has halved.
Show that A = 1200pi (12 - t).
(You may assume that the surface area of a sphere is given by A = 4pi r^2 , where r is
Well if you integrate the equation you get an equation for A in terms of t.

Now you don't know the value of k or the initial surface area of the sphere, which is why you are given its initial radius and the value of its radius after a certain amount of time.
11. Re: Writing differential equations, C4.
(Original post by Zishi)
You've to solve the differential equation by taking integral of both sides(after you take dt on right hand side, of course). What you'll get after that?
Is it

A = 1/t( -k^t )
12. Re: Writing differential equations, C4.
(Original post by hash007)
Is it

A = 1/t( -k^t )
No, when you integrate -k dt, it will give -kt + c.
So A = -kt + c, where c is another constant.
Last edited by Zishi; 04-06-2012 at 13:23.