An ice cream cone is left sitting in the hot sun. Sarah notices that the ice cream melts and loses half of its volume every 5 minutes. If the starting volume was 125 mL, determine a function for the volume, with respect to the amount of time left out in the sun.

Question

anonymous · Answer

Hi all. I have no idea how to do this.

imqwerty · Answer

hey robert

anonymous · Answer

Hey qwerty

anonymous · Answer

Long time no see

imqwerty · Answer

hahaha :D welcome back

anonymous · Answer

Thanks bro

anonymous · Answer

I'm traveling through Venezuela now

imqwerty · Answer

cool :D
ok so the volume is expanding 
it will be given like this-$$V=V_o(1+\gamma \Delta T)$$here 
γ=coefficient of 3D expansion
V_o is the initial volume
V=final volume

anonymous · Answer

Oh ok.

anonymous · Answer

Let me see if I understand the equation

imqwerty · Answer

i am not so sure i am also currently studying this topic :) but i think its ok
I have to leave now i have to go somewhere

baru · Answer

i think its exponential decay

anonymous · Answer

It is an exponential decay

baru · Answer

5 min is the half life

anonymous · Answer

Correct. But it's been a long while since I did exponential decay

anonymous · Answer

I think Qwerty's solution seems solid but that equation needs to be substituted with values I believe.

imqwerty · Answer

is this related to solid dynamics?

anonymous · Answer

It's related to advanced functions.

baru · Answer

i first came across it in context of radioactive elements

anonymous · Answer

And I'm sure physics explains better but this is some math jam

anonymous · Answer

How may I express every five minutes part in exponential decay equation? My skills are decayed.

imqwerty · Answer

wait a sec ima attach some of my notes  related to this

anonymous · Answer

Oh Thanks bro

ganeshie8 · Answer

Try :
$$V(t) = 125*(1/2)^{t/5}$$

anonymous · Answer

That sums up everything....

ganeshie8 · Answer

when t=5, we get : 

V(5) = 125*(1/2)^1 = 125/2

so the volume does get halved and we're good

anonymous · Answer

Wow @ganeshie8  how can you think of math like a language?

ganeshie8 · Answer

its easy to guess that function by trial and error haha

baru · Answer

@ganeshie8 thats very interesting!!
why is it that radioactive decay is modelled by an exponential function?

anonymous · Answer

Ahh I see. I really need to get the differential equation going.

ganeshie8 · Answer

They are same thing.
We could also solve a differential equation and get the same answer. The rate of change of volume is proportional to the existing volume. It is -1/2V per 5 minutes, which is same as -1/10V per minute :
$$\dfrac{dV}{dt} = -\dfrac{V}{10}$$

baru · Answer

they are the samething?
both the exponential funct and yours solve the differential equation?

anonymous · Answer

I think there are many ways of going about this problem.

anonymous · Answer

The one Ganesha introduced me at the beginning is more elementary

ganeshie8 · Answer

$$2^r = e^{r*\ln 2}$$

anonymous · Answer

And one with differential equation is more advanced and uses mathematical logics.

anonymous · Answer

Scott wants to calculate the distance from his house to each of his friends' houses. If he drives at 50 km/h, find a function for the distance, with respect to the number of hours it takes to travel.

anonymous · Answer

I think this can be expressed using differential equations too.

anonymous · Answer

F(x)=50km/h* x(h)

anonymous · Answer

Where x is the over all distance and oopps, the y being the hours taken

anonymous · Answer

But let's see how we can express this in differential equation.

ganeshie8 · Answer

that looks good, but we normally use "t" for time..

ganeshie8 · Answer

d(t) = 50km/h* t

anonymous · Answer

ah ok.. Sorry for my rusty math.... what a shame.

ganeshie8 · Answer

solving below differential equation gives the same function 
$$\dfrac{ds}{dt} = 50$$

anonymous · Answer

Ah ok that rings my bell.

anonymous · Answer

Thanks so much Ganesha!

anonymous · Answer

Math is wonderful thing you know

baru · Answer

$$e^{-kt}=(1/2)^{t/5}$$

??

ganeshie8 · Answer

Yes, 
$$k= \ln(1/2)/5$$

baru · Answer

phew!!
so all is ok with the world
thanks!!!

ganeshie8 · Answer

haha my previous reply has a mistake https://s3.amazonaws.com/upload.screenshot.co/8f51321707

ganeshie8 · Answer

v/10 is not a continuous change
so we need to massage this expression further while setting up the de

baru · Answer

ohh..you mean you've got the rate constant wrong k=1/10 wrong?

ganeshie8 · Answer

yes, 

$V(t) = 125e^{-kt} ~~$

we may use $V(5) = 125/2$ to solve $k$

ganeshie8 · Answer

the differential equation should be simply 
$$\dfrac{dV}{dt} = -kV$$

$k$ is a constant which can be worked from the given info

baru · Answer

yes :)
thanks again

ganeshie8 · Answer

np:)