In a 60km journey going up hill the upward slop slowed down a car.due to the slowdown, the average speed of car reduce by 15km/h and the time of journey increased by 20 minutes.what is duration of the journey?

To determine the duration of the journey, we can break down the problem using the information provided. Let's start by defining some variables and using the relationships between speed, distance, and time.

Setting Up the Problem

We know the following:

• The distance of the journey is 60 km.
• The average speed of the car decreases by 15 km/h due to the uphill slope.
• The time taken for the journey increases by 20 minutes.

Defining Variables

Let’s denote:

• v = original average speed of the car (in km/h)
• t = original time taken for the journey (in hours)

Relating Speed, Distance, and Time

From the basic formula of speed, we have:

Distance = Speed × Time

For the original journey, we can express the time as:

t = 60 / v

Adjusting for the Slowdown

With the slowdown, the new average speed becomes:

v - 15

The new time taken for the journey, which is 20 minutes longer than the original time, can be expressed as:

t + 1/3 (since 20 minutes is 1/3 of an hour).

Setting Up the Equation

Now, we can set up the equation for the new journey:

60 = (v - 15) × (t + 1/3)

Substituting the expression for t from earlier:

60 = (v - 15) × (60/v + 1/3)

Solving the Equation

Expanding this equation gives:

60 = (v - 15) × (60/v + 1/3)

Multiplying both sides by v to eliminate the fraction:

60v = (v - 15)(60 + v/3)

Distributing on the right side:

60v = 60v - 900 + v^2/3 - 15v

Rearranging gives:

0 = v^2/3 - 15v - 900

Multiplying through by 3 to eliminate the fraction:

0 = v^2 - 45v - 2700

Using the Quadratic Formula

Now, we can apply the quadratic formula:

v = [45 ± √(45² - 4 × 1 × (-2700))] / 2 × 1

Calculating the discriminant:

45² = 2025

4 × 2700 = 10800

2025 + 10800 = 12825

Now, substituting back into the formula:

v = [45 ± √12825] / 2

Calculating the square root of 12825 gives approximately 113.25:

v ≈ [45 ± 113.25] / 2

This results in two potential speeds, but only the positive speed makes sense in this context:

v ≈ 79.125 km/h

Finding the Duration

Now that we have the original speed, we can find the original time:

t = 60 / v ≈ 60 / 79.125 ≈ 0.758 hours

To convert this to minutes:

0.758 × 60 ≈ 45.5 minutes

Final Duration of the Journey

Thus, the duration of the journey, before the slowdown, is approximately 45.5 minutes. After accounting for the additional 20 minutes due to the uphill slope, the total time taken for the journey becomes:

45.5 + 20 = 65.5 minutes

In summary, the duration of the journey is approximately 65.5 minutes.