a search plane takes off from an airport at 6 am and travelsnorth at 200 miles p

Question

a search plane takes off from an airport at 6 am and travelsnorth at 200 miles per hour. a second plane takes off 6 30 am andtravels east at 170 miles per hour. the planes carry radios with amaximum range of 500 miles. when to the nearest minute will theseplanes no longet be able to communicate with each other please show how a search plane takes off from an airport at 6 am and travelsnorth at 200 miles per hour. a second plane takes off 6 30 am andtravels east at 170 miles per hour. the planes carry radios with amaximum range of 500 miles. when to the nearest minute will theseplanes no longet be able to communicate with each other please show how

Explanation / Answer

Analysis: -the first plane travels North or up at 200mph at 6am -30minutes or .5 an hour later the second plane leaves east orright at 170mph -the distance between the two planes is the hypotenus of aright angle triangle (forgive my spelling) -the distance formula between these two points on the triangleis a2 + b2 = c2 -the distance each individual plane can calculated by theformula (rate x time = distance --> r x t = d) Solution: -first you need to calculate a formula for the distancetraveled each plane takes by using the formula (rate x time =distance) -the rate of the first plane is 200mph so.... 200 x time = distance --> 200 x t = d equivalentexpression for simplicity --> 200t = d for example if you plug in t=0 you will a distance of 0miles (makes sense so far!) -the distance of the second plane can be calculated exactlythe same using r x t = d -the rate of the second plane is 170mph so... 170 x time = distance --> 170t = d -BUT the second plane took off 30 minutes after the firstplane or .5 (half) an hour -So how far is the first plane when the second plane takesoff? -lets find out using the first's planes distance formula 200t= d plug in .5 for t because time has to be in hours becauseyour rate is miles per hour (m/h) so (m/h) x h = m (justmiles!) anyways 200 x .5 = 100miles! so the first plane is 100 miles away already when thesecond plane leaves -to compensate for this distance you have to add 100 miles tothe first planes distance formula so you get this.... 200t + 100 = d this is the corrected formula for this situation right!plug in 0 hours for time and you get 100 miles! -lets check if this agrees with the second planes distanceformula, so lets plug in 0 hours for the second planes'sdistance formula 170 x 0 = 0miles (yes this agrees because theplane hasnt traveled at all yet while the first plane alreadytraveled 100miles as we saw with the first formula) -Finally the two equations we just found are the legs of theright angle triangle right!? yes! -so lets use a2 + b2 =c2 -the first plane is part a of the triangle and the secondplane is part b of the triangle so.... a = 200t + 100 and.... b = 170t and we have to find out when the distance is 500milesequivalent to saying when c = 500 so... a2 + b2 = c2 -->(200t + 100)2 + (170t)2 = 5002 -ok ok I know its long , but you said you wanted tosee how -now we can solve for t which will give us our time that weneed in hours then we will convert this to minutes afterwards -factor a2 which is (40000t2 + 40000t +10000) - factor b squared whichis (28900t2) - factor c squared which is 250000 so.. (40000t2 + 40000t + 10000) +(28900t2) = 250000   --> a2 +b2 = c2 -simple algebra simplification gives you 68900t2 + 40000t - 240000 = 0 -ok I did the hard part now use the quadratic formula to solvefor t which in this case is t= -40000 + ((400002 -( 4 x 68900 x 240000))/ 2 x 68900) after you get that time multiply it by 60 to get it intominutes and round to the nearest minute -ok ok I know its long , but you said you wanted tosee how -now we can solve for t which will give us our time that weneed in hours then we will convert this to minutes afterwards -factor a2 which is (40000t2 + 40000t +10000) - factor b squared whichis (28900t2) - factor c squared which is 250000 so.. (40000t2 + 40000t + 10000) +(28900t2) = 250000   --> a2 +b2 = c2 -simple algebra simplification gives you 68900t2 + 40000t - 240000 = 0 -ok I did the hard part now use the quadratic formula to solvefor t which in this case is t= -40000 + ((400002 -( 4 x 68900 x 240000))/ 2 x 68900) after you get that time multiply it by 60 to get it intominutes and round to the nearest minute

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