Kunni wrote:
ScottTargetTestPrep wrote:
Bunuel wrote:
Cities A and B are in different time zones. A is located 3000 km east of B. The table below describes the schedule of an airline operating non-stop flights between A and B. All the times indicated are local and on the same day.
Assume that planes cruise at the same speed in both directions. However, the effective speed is influenced by a steady wind blowing from east to west at 50 km per hour.
What is the time difference between A and B?
(A) 1 hour
(B) 1.5 hours
(C) 2 hours
(D) 2.5 hours
(E) Cannot be determined
Are You Up For the Challenge: 700 Level QuestionsAttachment:
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Let the time difference between cities A and B be x and the cruising speed of the plane be r. Since the wind speed is 50 km per hour blowing from east to west. The effective speed of the plane going from A to B is r + 50 and that going from B to A is r - 50.
Therefore, we can create the equations:
Goring from A to B: 3000/(r + 50) - x = 4 (notice that 4 is the no. of hours between 4 pm and 8 pm)
Goring from B to A: 3000/(r - 50) + x = 7 (notice that 7 is the no. of hours between 8 am and 3 pm)
If we add these two equations together, we have:
3000/(r + 50) + 3000/(r - 50) = 11
Multiplying the above equation by (r + 50)(r - 50) = r^2 - 2500, we have:
3000(r - 50) + 3000(r + 50) = 11(r^2 - 2500)
3000r - 150,000 + 3000r + 150,000 = 11r^2 - 27,500
6000r = 11r^2 - 27,500
11r^2 - 6000r - 27,500 = 0
(11r + 50)(r - 550) = 0
r = -50/11 or r = 550
Since r can’t be negative, r = 550. Substituting this for r into one of the two original equations (say the first one), we have:
3000/(550 + 50) - x = 4
3000/600 - x = 4
5 - x = 4
1 = x
Answer: ADid you calculate the roots by
-b +- Square root (b square - 4*ac)/2a ??
Reply to Kunni: Using the quadratic formula is one way to obtain the roots; however, with such large numbers, the calculations involved would have been nasty.
Instead, I observed that 11r^2 can only be factored as 11r * r; so I wrote:
11r^2 - 6000r - 27,500 = (11r - a)(r - b)
Opening up the parentheses, I obtained:
11r^2 - 6000r - 27,500 = 11r^2 - (a + 11b)r + ab
So, the product of the roots is -27,500 and the roots satisfy a + 11b = 6000. I simply looked for values where the product is -27,500 and which satisfies the equation a + 11b = 6000. It takes a few tries, but I think it is faster to obtain the roots this way compared to the quadratic equation.
Alternate solution: When factoring a quadratic equation of the form ax^2 + bx + c = 0 where a ≠ 1, you can always “convert” it to one that is a = 1 by removing it (from x^2) and multiplying it with c, the constant term c. For example, 2x^2 - x - 3 = 0 becomes x^2 - x - 6 = 0. Then we factor x^2 - x - 6 = 0 (i.e., the transformed equation) instead. Of course, after the roots of x^2 - x - 6 = 0 are found, we have to modify them so that they can be the roots of 2x^2 - x - 3 = 0. This is how:
x^2 - x - 6 = 0
(x + 2)(x - 3) = 0
x = -2 or x = 3
Now, here is the adjustment: for the two roots found we divide each by 2 (i.e., the value of a): -2/2 = -1 and 3/2.
These two new values will be the roots of the original equation. We can verify them by factoring 2x^2 - x - 3 = 0 directly:
2x^2 - x - 3 = 0
(2x - 3)(x + 1) = 0
2x - 3 = 0 → x = 3/2
or
x + 1 = 0 → x = -1
Now, back to the equation 11r^2 - 6000r - 27,500 = 0. Since 11 x 27,500 = 302,500, so the transformed equation we are going to factor instead is r^2 - 6000r - 302,500 = 0. Although 302,500 is the big number, recall one thing about factor quadratic equation with 1 as the coefficient of x^2 (or in this case, r^2) is: If constant term is negative, we look for two numbers whose product is (the absolute value of) the constant term and whose difference is (the absolute value of) the coefficient of x (or in this case, r). So we are looking for two numbers whose product is 302,500 and whose difference is 6000. That is, the two numbers must be one large and the other small. Actually, the large number must be around 6000 if the small number, say, is less than 100. Notice that 302,500/6000 is about 50 and since 50 divides into 302,500, one can guess the small number must be 50, which makes the large number to be 302,500/50 = 6050. We see that the difference between 6050 and 50 is exactly 6000, so we have found our two numbers. Now we can factor 11r^2 - 6000r - 27,500 = 0 as:
(r - 6050)(r + 50) = 0
r = 6050 or r = -50
Dividing both numbers by 11, we have 6050/11 = 550 and -50/11 as the roots of 11r^2 - 6000r - 27,500 = 0.
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