The volume of an ‘ice cream cone’ is that of the hemisphere, 2/3 πr3, plus that of the cone, πr2h/3. If the angle at the bottom of the cone is 30°, then we can make a right triangle with angle 15°. It’s height is r/tan(15°), or 3.372r. This makes the volume of the ice cream cone 5.372rπr3/3. This is 500ml, or 500cm3. We can then make the equation 5.372rπr2/3=500cm3, which solves to r=9.428. Now, the surface area of the hemisphere is 3πr2, and that of the cone, πr(r+√(h2+r2)). But we must subtract 2πr2 for the 2 circles that are not exposed. That leaves us with 2πr2 + πr√(h2+r2). Plug in r=9.428 and h=3.372r to get our answer. 1601cm2.
V=500cm^3=1.791\pi r^2\\[8pt] r=9.428cm\\[16pt] SA=2\pi r^2+\pi r\sqrt{h^2+r^2}\\[8pt] SA=2\pi r^2+\pi r\sqrt{1.07177r^2}\\[8pt] SA=5.732\pi r^2\\[8pt] SA=1601cm^2Day: October 20, 2019
Solution to Fascinating Frequencies 6
We’re trying to figure out the minimum amount of fuel that needs to be burned in order to bring a rocket back into geostationary orbit. Firstly, how much energy is released by the explosion? Sodium is an Alkali with atomic weight 23, and Fluorine is a Halogen with atomic weight 19. This means they are in the perfect proportions to explode. There is no precise amount of energy this explosion will produce. However, we’re trying to find how much fuel is necessary to ensure the scientists can get back to orbit. So we take the maximum. The first ionization energy of sodium is 4.26 eV. The electronegativity of fluorine is 3.98 eV. 4.26-3.98=0.28 electron volts of energy released per reaction. There are 6.022×1025 of each. The total energy released is 2.702×106 m2kg/s2 of energy. This produces a force of 2.702×103 metres kilograms per second. Divide this by the mass of the station, 500Mg, and you get a total acceleration of 5.404 ×10-3 metres per second of delta v. The most fuel efficient way to undo this is to simply wait until the rocket comes back and then switch back. This is because the rocket’s new orbit will still include the spot it changed orbit’s at, and in the case of prograde burns, this will be it’s perigee. The perigee also happens to be the time when retrograde burns are most efficient. The release of 5.404 ×10-3 metres per second of delta v requires a push of, guess, 2.702×103 metres kilograms per second of force, same as last time. This requires, again, 2.702×106 m2kg/s2 of energy. The liquid fuel used by the scientists, oxygen and hydrogen, releases 1.418 m2/s2. This does not seem to be the correct unit, it is missing the mass. That is because the mass is the unknown. We divide the total energy by this to find the total mass, 1.906×106 kilograms. But this weighs far more than the station! The scientists do not have enough fuel to get back into orbit.