master_q 0 Posted May 17, 2003 These two questions have still not been answered. There have been attempts, but still no one has answered them correctly. If no one gets them correct by Saturday, I’ll give the answer to both with an explanation. . . . . {Official} Star Trek Trivia #45******************* Quark Needs YOUR help! Quark somehow has time traveled to the past on Earth and is stuck! (He is rescued two years later and goes back to the 24th Century. When he was returned he was returned the exact time of which he traveled back through time) The cost of fuel to run a locomotive is proportional to the square of the speed and $25 per hour for a speed of 25 miles per hour. Other costs amount to $100 per hour, regardless of the speed. Find the speed that minimizes the cost per mile for Quark! ******************* ??? Answering: Send myself a private message (do not answer here!) Points: 15 Hints: It is not as hard as it looks. Just think it out logically. Write out the important facts on a piece of paper (that’s what I do when solving a problem like this one) and then once you have it down on a piece of paper I think (and hope) that you might be able to figure it out. Let v be the required speed and let C be the total cost per mile Fuel cost per hour is kv^2 Where k is the constant term that you are going to have to figure out (it is an unknown) Good Luck! {Official} Star Trek Trivia #49 ******************* Spock Needs YOUR help! Spock in an experiment is using a steam-jacketed eudiometer and filled it with 32 mL of oxygen and 4 mL of hydrogen over mercury. How much of what gas would be left uncombined after sparking? ******************* A. None of either B. 3 mL H2 C. 24 mL O2 D. 28 mL O2 E. 30 mL O2 F. 55 mL O2 G. None of the above options Answering: Send myself a private message (do not answer here!) Points: 15 Good Luck!! Master Q StarTrek_Master_Q@yahoo.com Share this post Link to post Share on other sites
master_q 0 Posted May 17, 2003 #45 No one got #45 correct :-( However, I’ll give the answer It is 50 miles per hour! “Why?” you ask . . . . . It is pretty easy once you use some good old basic calculus. We know this: Let v be the required speed and let C be the total cost per mile Fuel cost per hour is kv^2 So C = (cost in $/hour) / (speed in miles/hour) = { (v^2)/25 +10 }/v = v/25 + 100/v Then, (dC)/(dv) = 1/25 - 100/v^2 = { (v - 50)(v + 50) } / (25v^2) Because v>0, the only real # that is important is v = 50. This is because d^2C/dv^2 = 200/V^3 > 0. When v = 50 our second derivative test shows that C has a relative minimum at v = 50. So v = 50 and therefore the important # (or if you want to call it “critical #”) in (0, + infinity). C has an absolute minimum at v = 50. So the cut the costs the speed would be 50 miles per hour. I can hear a few people complaining of why C ‘s minimal would imply this minimal value would produce the least cost because we have this also to worry about. This in fact can be figured or reasoned out by thinking of a continuous function which we can define as interval j (where j can be finite or infinite). So if the function is a extremism within j, then the relative extremism is also an absolute extremum on j. Master Q StarTrek_Master_Q@yahoo.com Share this post Link to post Share on other sites
master_q 0 Posted May 17, 2003 #49 Well I would like to congratulate Indy in getting this correct! Indy got 15 points for giving the correct answer The answer is 30 mL O2!! This one is easier then the last because we don’t have to use calculus in solving it. H2 to O2 ratio by volume is 2:1 in the formation of water. So 4 mL H2 will react with 2 mL of O2 to make 4 mL of steam. 2H2 (g) + O2 (g) --> 2H2O (g) This leaves 30 mL of O2 uncombined Master Q StarTrek_Master_Q@yahoo.com Share this post Link to post Share on other sites
