Poor man's arbitrage

You've found a stockpile of 1955 United States pennies, which are now worth approximately twice as much as raw metal compared to their value as currency. A Fresnel lens can be used to concentrate incoming light onto a focal point.

For how many seconds would you need to focus the Fresnel lens on a penny in order to melt the entire coin?

Details and assumptions:

  • A penny weighs \(2.5 \text{ g}\) and is made of \(100\)% \(\text{Cu}.\)
  • The penny starts out at \(25^\circ\text{C}\) and melts at \(1085^\circ\text{C}.\)
  • \(\text{Cu}\)'s heat of melting is \(176 \text{ kJ}/\text{kg}\) and its specific heat is \(0.386\text{ kJ}/\text{kg}\cdot\text{K}.\)
  • The area of the lens is \(0.2 \text{ m}^2,\) the power of the sun is \(1370\text{ W/m}^2,\) and \(100\)% of the energy goes into heating the coin.

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