A certain vehicle is undergoing performance adjustments during a set of three trials on an 8-mile track. The first time it travels around the track at a constant rate of p miles per minute, the second time at a constant rate of p^2 miles per minute, and the third time at a constant rate of p^3 miles per minute. If the vehicle takes the same time to travel the first 18 miles of these trials as it does to travel the last 8 miles, how many minutes does it take to complete all three trials (not counting any time between trials)? Select the correct option:
A) 18
B) 36
C) 60
D) 96
E) 112 minutes
The first journey takes 18 miles whereas the last journey takes 8 miles.   r1= p   r2=p^2   and r3= p^3 so we set up a table like   r x t = d   p x t1 = 8   p^2 x t2= 8   p^3 x t3 = 2 (because 18-16)   so the time taken to complete each leg individually amounts to t1 + t2 + t3 = (8/p) + (8/p^2) + (2/p^3). ----&gt; (equation 1)   On the other hand, the time taken to complete the last leg of miles is (8/p^3) ---&gt; (equation 2) because p^3 is the speed on the last leg. Since both the times are equal, equate them to get:   (8/p) + (8/p^2) + (2/p^3) = (8/p^3)   Taking the LCM of the Left hand equation and solving both these equations gives us the value of p=1/2 and p= (-3/2) Since p can not be negative, we have p=1/2 as our value.   Now put this value in the time taken to finish each trial which is the sume of (8/p) + (8/p^2) + (8/p^3) and it will become   16+32+64= 
<gadget id="calculator">  16+32+64</gadget>
<output>112</output>
112 minutes.   Option 112 minutes is the correct answer.
<result>E</result>
A certain vehicle is undergoing performance adjustments during a set of three trials on an 8-mile track. The first time it travels around the track at a constant rate of p miles per minute, the second time at a constant rate of p^2 miles per minute, and the third time at a constant rate of p^3 miles per minute. If the vehicle takes the same time to travel the first 18 miles of these trials as it does to travel the last 8 miles, how many minutes does it take to complete all three trials (not counting any time between trials)?