1.) If the average number of vehicles admitted through the gate of an amusement
ID: 3033266 • Letter: 1
Question
1.) If the average number of vehicles admitted through the gate of an amusement park per minute is equal to x, then the average waiting time in minutes for each vehicle at the gate can be computed by f(x)=x-5/x^2-10x where x>10. Estimate the admittance rate X that results in an average wait of 6 seconds.
2.) A parking garage attendant can wait on 39 cars per hour. If cars arrive randomly at a rate of x cars per hour, then the average line length is given by f(x)=x^2/1521-39x, where x-values are limited to 0
(a)Solve the inequality f(x)<12
Explanation / Answer
If the average wait is 6 seconds, then f(x) = (x-5)/(x2 -10x) = 6 or, x – 5= 6(x2 -10x) or, x -5 = 6x2 -60x or, 6x2 -60x –x +5 = 0 or, 6x2 -61x +5 = 0. On using the quadratic formula, we have x = [61 ± { (-61)2 -4*6*5}]/2*6 = [59± (3721-120)]/12 = (59± 3601)/12 = (59± 60)/12 (approx.) = (59+60)/12 ( as x cannot be negative) or, x = 119/12 = 9.92 or, say x = 10 ( on rounding off to the nearest whole number.
2. If f(x)< 12, then x2/(1521-39x) < 12 or, x2 < 12(1521-39x) or, x2 < 18252 – 468x or, x2 +468x -18252 < 0 or, x2 + 2*234 x +(234)2 – ( 18252+ 2342) < 0 or, ( x+234)2 < 18252 + 54756 or,( x+234)2 < 73008 . Thus, either x+234 < 73008 or, x+234 < 73008 = 270, or , x < 270-234 i.e. x < 36.
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