# calculas

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?

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