ice cube melting rate problem

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emilyf

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Anyone up for this challenge? I realy need help and don't know where to begin.

How long will it take an ice cube to melt completely? You left out an ice cube ant 1/4 of it melts in 1 hour. Use the following steps to find out how long it will take before it is melted completely.

1-let s represent the side length of the ice cube, and observe that s is a function of time t. Write expressions which describe the volume V and surface area A of the cube as functions of s (which you should observe, in turn makes them functions of t).

2-melting takes place at the surface of the cube, so you decided that it is reasonable to assume that the cubes volumn V decreases at a rate that is proportional to its surface area A. saying that a is proportional to b means that there is constant k so that a=kb.) write an equation that describes dV/dt in terms of s. Is your constant k a positive or negative number?

3-use the chain rule and your answer from #1 to write an expression which relates dV/dt to ds/dt

4-now you have two expressions (your answers from 2 and 3 for dV/dt).set them equal to find ds/dt in terms of the constant k.

5-use your answer from #4 to write an equation which relates s0=s(0) to s1=s(1), where t is measured in hours, and use it to find tmelt (the melting time) in terms of the quantity s1/s0.

6-taking into account that 1/4 of the cube melted in 1 hour, find an approximation to s1/s0 using the function you wrote in #1 for V in terms of s.

7-find tmelt. how much longer will you have to wait for the ice cube to melt completely
 
emilyf said:
Anyone up for this challenge? I realy need help and don't know where to begin.

How long will it take an ice cube to melt completely? You left out an ice cube ant 1/4 of it melts in 1 hour. Use the following steps to find out how long it will take before it is melted completely.

1-let s represent the side length of the ice cube, and observe that s is a function of time t. Write expressions which describe the volume V and surface area A of the cube as functions of s (which you should observe, in turn makes them functions of t).

2-melting takes place at the surface of the cube, so you decided that it is reasonable to assume that the cubes volumn V decreases at a rate that is proportional to its surface area A. saying that a is proportional to b means that there is constant k so that a=kb.) write an equation that describes dV/dt in terms of s. Is your constant k a positive or negative number?

3-use the chain rule and your answer from #1 to write an expression which relates dV/dt to ds/dt

4-now you have two expressions (your answers from 2 and 3 for dV/dt).set them equal to find ds/dt in terms of the constant k.

5-use your answer from #4 to write an equation which relates s0=s(0) to s1=s(1), where t is measured in hours, and use it to find tmelt (the melting time) in terms of the quantity s1/s0.

6-taking into account that 1/4 of the cube melted in 1 hour, find an approximation to s1/s0 using the function you wrote in #1 for V in terms of s.

7-find tmelt. how much longer will you have to wait for the ice cube to melt completely
Click to expand...

We are up for the challenge - but you need to show your work .................

Looks like you are putting up mid-term exam problems here.
 
I just can't figure out where to begin. It is a take home practice question and we can use any help available to us. I will keep working on it but would still appreciate any ideas.Thanks
 
The problem gives you excellent hints - start with part (1) - Draw a sketch

each side is x.

What is the total surface(A) of a cube in terms of x?

Now find the inverse - what is the expression for side (x) interms of area (A)? ....................... (1)

What is the total Volume (V) of a cube in terms of x?

Now using (1) - What is the total Volume (V) of a cube in terms of A?
 
thanks-that was very helpful. I will keep working on it and let you know what I come up with.
 
total surface area in terms of x is 6x, inverse is 1/6x (?)
volumn of a cube in terms of x is xcubed
total volumn of the cube in terms of surface area is 1/6cubed?
 
emilyf said:
total surface area in terms of x is 6x, inverse is 1/6x (?)
volumn of a cube in terms of x is xcubed
total volumn of the cube in terms of surface area is 1/6cubed?
Click to expand...

Each EDGE of the cube has a length of x units. Each FACE of the cube is a square, with sides x units long. What is the area of one face?

A cube has 6 faces, so the total surface area will be 6 * (area of one face).

You need to make a bit of correction in the surface area before you proceed further.
 
area of 1 side is xsquared, so area of the cube is 6xsquared?
I'm sorry-I really don't get this question at all. We didn't learn much (if any) of this yet-my teacher wants us to see if we can figure it out on our own, and I'm at a loss.
 
emilyf said:
area of 1 side is xsquared, so area of the cube is 6xsquared?
I'm sorry-I really don't get this question at all. We didn't learn much (if any) of this yet-my teacher wants us to see if we can figure it out on our own, and I'm at a loss.
Click to expand...

I re-checked the information in the problem...it says that the length of the side of the cube starts out as s units.

S0, if the length of each edge of a cube is s units, then the area of one side would be s[sup:1i4ek3ao]2[/sup:1i4ek3ao] square units and the total surface area of the cube is 6s[sup:1i4ek3ao]2[/sup:1i4ek3ao] square units.

The original volume of the ice cube would be s*s*s, or s[sup:1i4ek3ao]3[/sup:1i4ek3ao] units cubed.

You posted this in the calculus category...surface area and volume of a solid like a cube are usually covered in geometry.

Since calculus isn't really my strong point, I'll leave the rest of the parts of your question to someone more knowledgeable than I.
 
thanks so much for your help-the rest is calculas and I am really at a loss since we have not learned this yet. hopefully someone can work it out for me so I can learn it!
 
emilyf said:
thanks so much for your help-the rest is calculas and I am really at a loss since we have not learned this yet. hopefully someone can work it out for me so I can learn it!
Click to expand...

So what is "your" answer to question #1?

A(s) = ???

V(s) = ???
 
emilyf said:
area= 6s squared
volumn= s cubed
Click to expand...

\(\displaystyle V \ = \ s^3\)

and

\(\displaystyle A \ = \ 6s^2\)

Very good - now you are given

\(\displaystyle \frac{dV}{dt} \ = \ -k \cdot A\)

Now what do we do......
 
substitute numbers? d(scubed)/dt=-k(6ssquared)

not sure at all if this is right or what I missed
 
2-melting takes place at the surface of the cube, so you decided that it is reasonable to assume that the cubes volumn V decreases at a rate that is proportional to its surface area A. saying that a is proportional to b means that there is constant k so that a=kb.)


write an equation that describes dV/dt in terms of s. Is your constant k a positive or negative number?


\(\displaystyle \frac{dV}{dt} \ = \ -k*6*s^2\) .................you got that!

Now tackle problem #3
 
we've just barely touched on the chain rule in class-I will give it a shot but this will probably not be right

dv1/dt=(-k times 6 times ssquared) times (k+6x)

I know that's wrong-but it's the best shot I can give it with what we've learned so far. Please explain the next step to me.
 
Could someone please keep helping me with this problem? I don't know how to finish it and I need to figure it out by this weekend. Thanks! All the helpers are so great :D
 
Using chain rule:

\(\displaystyle \frac{dV}{dt} \ = \ \frac{dV}{ds} \ * \ \frac{ds}{dt}\)

You know:

V = s[sup:3w5dhjo6]3[/sup:3w5dhjo6]

What do you get for:

\(\displaystyle \frac{dV}{ds}\)
 
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