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18_04 THINKING CAP ANSWERSPuzzle 1. [PRIZE PUZZLE] A dome on MarsThe dome should have a height of 117.46 metres. A dome is geometrically a spherical cap. The standard formula for the volume of a spherical cap is V = ((Pi h^2)/3) * (3r –h)where Pi is the constant 3.141592654 etc, h is the unknown height of the cap, ^2 means squared,* means multiplied,r is the radius of the sphereNote that r is the unknown radius of the sphere, not the known radius (150m) of the base of the cap (or dome), which we shall call b. Unless the dome is a hemisphere, b is smaller than r. So the formula needs to be rewritten in terms of b.549275118745rbh00rbh647700-952500549275111760r00r6591303746500Draw a right-angled triangle whose hypotenuse r is the radius of the sphere, and whose leg b is the radius of the base of the cap. From Pythagoras’ Theorem, r^2 = b^2 + (r-h)^2 r = (b^2 + h^2) / 2hSubstitute for r in the original formula, and after some simplification:V = 1/6 * Pi h * (3b^2 + h^2)We have been told the diameter of the base of the dome is 300m, therefore b = 150m.Solving for b = 150, and V = 5,000,000h = 117.46 Puzzle 2. A debtor's defenceHis argument was incorrect. Though some cells are replaced quickly, others, such as neurons, are very long-lasting.Nevertheless, in English law he might not have to pay, because under the Limitations Act 1980 there is a time limit of six years for action for recovery in simple contracts.Puzzle 3. The diminishing sum14 years ago.1,000,000 * (EXP (-0.05 * 14)) = 496,585Puzzle 4. True or False?True, according to researchers at the University of California San Francisco, who studying the living mouse lung observed that many blood platelets were produced. It had previously been assumed the components of blood were produced mainly by bone marrow. ................
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