On a journey of 200km a motorist found out that if he were to increase his speed by 2km/h the journey will take 15 minutes less calculate his average speed
Let's say the motorist's average speed for the journey is "x" km/h.
If he were to increase his speed by 2 km/h, his new speed would be "x + 2" km/h.
The journey distance is 200 km.
We'll first calculate the time taken for the journey at the average speed "x" km/h.
Time taken = Distance / Speed
= 200 km / x km/h
= 200 / x hours
Next, we'll calculate the time taken for the journey at the increased speed of "x + 2" km/h.
Time taken at increased speed = Distance / Speed
= 200 km / (x + 2) km/h
= 200 / (x + 2) hours
According to the problem, the journey will take 15 minutes less if the motorist increases his speed by 2 km/h. Since there are 60 minutes in an hour, 15 minutes is equal to 15/60 = 1/4 of an hour.
So, the difference in time taken for the journey at both speeds is 1/4 hour.
\(\frac{200}{x} - \frac{200}{x+2} = \frac{1}{4}\)
To simplify the equation, let's multiply through by 4(x)(x+2) to get rid of the fractions:
800(x+2) - 800x = x(x+2)
800x + 1600 - 800x = x² + 2x
1600 = x² + 2x
Rearranging the equation:
x² + 2x - 1600 = 0
Now, let's solve this quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Here, a = 1, b = 2, and c = -1600.
x = (-2 ± √((2)² - 4(1)(-1600))) / (2 * 1)
x = (-2 ± √(4 + 6400)) / 2
x = (-2 ± √(6404)) / 2
x = (-2 ± 80.02) / 2
Ignoring the negative value (as average speed cannot be negative), we have:
x = (80.02 - 2) / 2
x = 78.02 / 2
x = 39.01 km/h
Therefore, the motorist's average speed for the journey is approximately 39.01 km/h.
To solve this problem, we can set up an equation using the formula for average speed:
Average Speed = Total Distance / Total Time
Let's represent the motorist's original speed as "x" km/h and the time it takes for the journey at that speed as "t" hours.
According to the information given, increasing the speed by 2 km/h reduces the journey time by 15 minutes, which is equal to 15/60 = 1/4 hour.
So, at the increased speed of (x + 2) km/h, the journey would take (t - 1/4) hours.
Now, we can set up two equations based on the distance and time:
Equation 1: x * t = 200 (distance Equation)
Equation 2: (x + 2) * (t - 1/4) = 200 (distance Equation)
Let's solve these equations to find the value of average speed.
From Equation 1, we have:
xt = 200
t = 200 / x
Substituting this value of "t" into Equation 2:
(x + 2) * (200 / x - 1/4) = 200
(x + 2) * (800 - x) = 200 * 4
(x + 2) * (800 - x) = 800
Expanding the equation:
800x + 1600 - x^2 - 2x = 800
Rearranging terms and simplifying:
x^2 - 800x + 800 = 0
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Here, a = 1, b = -800, and c = 800.
Solving for x gives two solutions:
x = (800 ± √(800^2 - 4 * 1 * 800)) / (2 * 1)
x = (800 ± √(640000 - 3200)) / 2
x = (800 ± √(636800)) / 2
Calculating the square root:
x ≈ (800 ± 798.25) / 2
We have two possible values for x:
x1 ≈ (800 + 798.25) / 2 ≈ 1299.13 (ignoring the negative value as speed cannot be negative)
Now, we can substitute x into Equation 1 to find t:
t = 200 / x1
t ≈ 200 / 1299.13 ≈ 0.154
Finally, calculating the average speed:
Average Speed = Total Distance / Total Time
Average Speed = 200 / 0.154 ≈ 1298.70 km/h
Therefore, the average speed of the motorist for the journey of 200 km is approximately 1298.70 km/h.