A certain vehicle is undergoing performance adjustments during a set

Question

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)?

(A) 18
(B) 36
(C) 60
(D) 96
(E) 112

Kudos for a correct solution.

(A) 18
(B) 36
(C) 60
(D) 96
(E) 112

Kudos for a correct  solution .

Jerry1997 · Accepted Answer

1. Time taken for first 18 miles =  \frac{8}{p} + \frac{8}{p^2}+\frac{2}{p^3} 
2. For the last 8 miles = \frac{8}{p^3} 
As per the problem,
 \frac{8}{p} + \frac{8}{p^2}+\frac{2}{p^3} =  \frac{8}{p^3} 
Solving for p, 
p= \frac{1}{2}

Therefore, time taken for all the 3 trials = \frac{8}{(1/2)} +\frac{8}{(1/4)} +\frac{8}{(1/8)}  = 16 +32 + 64 = 112
Ans: E

umair4 · Answer

This is a rate-time-distance problem. We would be better off setting a rate-time-distance table.

The first journey takes 18 miles whereas the last journey takes 8 miles.

We are told that:
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). ----> (equation 1)

On the other hand, the time taken to complete the last leg of miles is (8/p^3) ---> (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 = 112 minutes.
Option E is the correct answer.

p.s: if you like the explanation, please give a kudos

Tapabrata18 · Answer

Hi, I had a doubt on this question. As per my understanding, it is said "If the vehicle takes the same time to travel  the first 18 miles of these trials " and then "the  last 8 miles " Adding the above two cases(18+8=26) but the total length is (8*3=24)miles. How is this possible. Please let me know

Bunuel · Answer

The first 18 miles consists of 8 miles of the first trial + 8 miles of the second trial + the first 2 miles of the third trial. The last 8 miles consists of 8 miles of the third trial. So, there is an overlap of 2 miles.

jnitish25 · Answer

As per the problem,
8/p+8/p^2+2/p^3= 8/p^3 
=>  8/p+8/p^2=6/p^3 
Reqd time is 8/p + 8/p^2 + 8/p^3 =>Substituting above value =>6/p^3 + 8/p^3 i.e  14/p^3 .
Only  112  is a multiple of 14.

menonrit · Answer

Hi,

Could you please show the working out of how you got p =1/2

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A certain vehicle is undergoing performance adjustments during a set

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A certain vehicle is undergoing performance adjustments during a set

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