Find the Average SpeedDate: 11/27/97 at 08:59:37 From: ALI HAIDER Subject: Quadratic Equations A motorist has to travel 160km. His average speed is 8km/h slower than he anticipated and he takes 1 hour longer than intended. Find his actual average speed. I am unable to create an equation from this problem. Date: 11/27/97 at 13:35:06 From: Doctor Luis Subject: Re: Quadratic Equations This problem isn't so bad if you keep track of what is happening physically, and ask yourself what it means. Speed is nothing more than the distance covered by something over a period of time. Mathematically, you express this relation as follows, distance covered avg speed = ------------------------------------ time it took to cover that distance Symbolically, you can write that as v = d / t Solving for the distance, d = vt So you see, the distance covered is the product of the speed times the time. That makes sense because the faster you travel the more distance you cover in the same amount of time. However, in this problem, the distance to be covered is the same. That means that if you travel faster (v is greater) you will get to where you want earlier (t is less). Similarly, if you travel slower (v is less), it will take more time to get where you want (t is greater). For example, to travel 160 km, you could travel at 80km/hr for 2 hours, or at 5 km/hr for 32 hours (1.33 days), but the product of the speed times the time remains the same (because it's the same distance). Now, the problem tells you that he expected to travel 160km at a speed v (we don't know what it is yet, so we just represent it by a letter) in a time of t hours. Since he traveled 8 km/hr slower than expected (v-8 km/hr), the problem tells us that it took him 1 hour longer (t+1 hours) to travel those same 160 km. This means that, for his trip anticipated trip actual trip d = vt d = (v-8)(t+1) 160 = vt 160 = vt - 8t + v - 8 Try to explain what the equations mean based on what I explained to you. (Remember, v was his anticipated speed, and t was his anticipated time) The two equations above imply, - 8t + v - 8 = 0 - 8(160/v) + v - 8 = 0 (because t = 160/v) v^2 - 8v - 1280 = 0 (multiplying both sides by v) (v - 40)(v + 32) = 0 (factoring) Now, either v - 40 = 0, or v + 32 = 0 which means that either v = 40 or v = -32 Since the negative solution doesn't make sense (why?), we conclude that he was expecting to travel at an average speed of 40 km/hr, so his actual speed is 8km/hr less, or 32 km/hr. I don't know how much math you've had, but if you have any questions whatsoever, or there is anything you didn't understand, ask again and we'll explain in more detail whatever it was you didn't get... By the way, can you find out how long it actually took him to travel those 160 km? -Doctor Luis, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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