Question: Points A and B are 120 km apart. A motorcyclist starts from A to B along straight road AB with speed 30 kmph. At the same time a cyclist starts from B along a road perpendicular to road AB, with a speed of 10 kmph. After how many hours will the distance between them be the least?

1. 3 hours
2. 3.4 hours
3. 3.5 hours
4. 3.6 hours
5. None

“Points A and B are 120 km apart. A motorcyclist starts from” - is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solutions and Explanation

Approach Solution : 1

The length of the hypotenuse would be the distance between two riders, which is equal to the square root of (120−30x)^2 + (10x)^2 = 1000x^2 − 60 ∗ 120^x + 120^2 (where x is the time in hours).

Therefore, the value of the quadratic expression must be minimized (function),
1000x^2 − 60 ∗ 120x + 120^2.

The minimum of the quadratic function f(x)=ax^2 + bx + c is now reached (or maximum when a is negative - not our case),
when x = −(b/2a) = (60∗120)/(2∗1000) = 3.6

Correct Answer: (D)

Approach Solution : 2

The first motorcycle rider covered a distance of 30 t in t time
The second motorcycle rider covered a distance of 10t miles in t seconds.

Now, whenever this happens, a right-angled triangle with the following sides is created.

1. The cycle rider's distance traveled
2. The remaining space between the motorcyclist and B
3. The hypotenuse is the current distance between the two riders.

As a result, the equation of their distance becomes

d^2 = (10t)^2 + (120-30t)^2

Which is maximum when its differential is 0

Therefore, 20t = 2(120-30t) *3

As a result, t = 3.6

Correct Answer: (D)

Approach Solution : 3

Finding the shortest hypotenuse in a triangle formed by a biker on road AB and a cyclist traveling perpendicular to AB from point B is the basic goal of this problem. We can use the graph to enter data into the quadratic formula to determine the length C that is the shortest.

a^2+b^2=c^2

In one hour, the cyclist has covered 10 km and the biker has covered 30 km.

90^2 + 10^2 = c^2
8200 = c^2

Finding the square is unnecessary because we can compare values of c^2.

In two hours, the cyclist has covered 20 km and the biker has covered 60 km.

60^2+20^ = c^2\s4000 = c^2

In three hours, the cyclist has covered 30 km, and the biker has covered 90 km.

30^2 + 30^2 = c^2\s1800 = c^2

The biker and cyclist get closer with each passing hour. Out of all the options, the fourth option that states 3.6 shortens the biker's time the most (and, consequently, the square of a faster speed, such as 30 km/h). It's also possible to simply glance at the graph and note that the red line between the biker and cyclist gets shorter as more time passes.

Correct Answer: (D)

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