Optimizing Travel Time: A Mathematical Analysis

Travel time optimization involves understanding the relationships between distance, speed, and time. This article delves into a specific scenario where two individuals, A and B, are traveling different distances at different speeds. We’ll analyze their travel times and speeds to determine how long it takes for B to cover a 90 km distance at the same speed as A’s initial conditions.

Understanding the Scenario

The problem at hand involves two individuals, A and B, traveling distances of 40 km and 80 km, respectively. The conditions are as follows:

A takes 2 hours more than B to cover 40 km. When A doubles his speed, he takes 1.5 hours more than B to cover 80 km.

Our goal is to determine how long it will take B to cover 90 km at the same speed as A's initial conditions.

Mathematical Representation

Let’s denote the speed of B as v km/h, and the time taken by B to cover 40 km as t hours. Therefore, the time taken by B to cover 40 km is given by:

t 40/v

Since A takes 2 hours more than B to cover 40 km, the time taken by A is:

tA t 2 40/v 2

Let’s denote the speed of A as u km/h. The time taken by A to cover 40 km is:

tA 40/u

Setting the two expressions for tA equal gives us:

40/u 40/v 2

Multiplying through by uv to eliminate the denominators yields:

40v 40u 2uv

Rearranging the equation:

2uv - 40u - 40v 0 (1)

Second Condition

Next we consider the second condition: if A doubles his speed, he will take 1.5 hours more than B to cover 80 km.

When A’s speed is doubled, his new speed is 2u km/h. The time taken by A at this speed to cover 80 km is:

tA, new 80/(2u) 40/u

The time taken by B to cover 80 km is:

tB 80/v

According to the problem, we have:

40/u 80/v - 1.5

Multiplying through by uv gives:

40v 80u - 1.5uv

Rearranging gives us:

1.5uv - 40v - 80u 0 (2)

Solving the Equations

Now we have two equations (1) and (2):

2uv - 40u - 40v 0

1.5uv - 40v - 80u 0

Lets solve equation (1) for u:

2uv - 40u - 40v 0

Rearranging gives:

2uv 40u - 40v implies u2v - 40 -40v implies u -40v/2v - 40

Substituting u into equation (2):

1.5vleft( -40v/(2v - 40) right) - 40v - 80left( -40v/(2v - 40) right) 0

Multiplying through by 2v - 40 to eliminate the denominator:

1.5( -40v) - 40v(2v - 40) - 80( -40v) 0

Expanding gives:

-60v2 - 80v2 1600v - 3200v 0

Combining like terms:

-140v2 - 1600v 0

Factoring out -20v:

-20v(7v 80) 0

This gives v 0 or v -80/7, which is not possible. Therefore, let’s directly solve for tB for a 90 km distance.

Determining Time for B to Cover 90 km

Time for B to cover 90 km is:

tB 90/v

Given the conditions, we assume v 20 or 30 for example:

If v 20, then tB 90/20 4.5 hours If v 30, then tB 90/30 3 hours

Considering the initial conditions, we can assume v 20 or 30 based on the problem conditions.

Conclusion

To cover 90 km, B will take:

tB 90/v hours, where v is determined from the initial conditions.

If further constraints or specific speeds are provided, we can refine the value of v for exact times.