What is the Minimum Altitude for Ice to Melt and the Best Engine Design?

[Farside]

I was running through some problems, doing just fine, until I came across these two. I can't work out a correct solution.

1) A 1-kg block of ice at 0 C falls into a lake whose water is also at 0 C, and 0.01 kg of the ice melts. What was the minimum altitude from which the ice fell?

2) Three designs for an engine to operate between 450 K and 300 K are proposed. Design A is claimed to require a heat input of 800 J for each 1000 J of work output, design B a heat input of 2500 J, and design C a heat input of 3500J. Which design would you choose and why?

 

[Doc Al]

Here are some hints that may help:
(1) How much energy is needed to melt that much ice? Where does that energy come from? (Consider gravitational potential energy.)
(2) What is the maximum possible efficiency of a heat engine operating between those two temperatures?

 

[Farside]

still can't get 'em

 

[Doc Al]

Show what you've done so far.

 

[Farside]

For the second, I'm assuming the max efficiency is about .33. So I would choose design B.

I have no clue of what to use for the first problem.

 

[Doc Al]

For the second, I'm assuming the max efficiency is about .33.

No need to assume it. Calculate it based on the temperatures.

So I would choose design B.

What is the stated efficiency of design B?

I have no clue of what to use for the first problem.

How much energy does it take to melt ice? Hint: $Q = m L_f$, where $L_f$ is the "latent heat of fusion" for water.