The distance between towns M and N is 280 km. A car and a bus travel from M to N.
The average speed of the bus is 20 km/h less than that of the car. The bus takes 1 h 10 min more than the car to travel from M and N.
(a) If the speed of the bus x km/h ?
car = x
bus x - 20 + 1 1/6
They said the speed of the bus is x.
time = distance/speed. So,
280/x = 280/(x+20) + 7/6
To solve this problem, let's start by setting up equations based on the given information.
Let the speed of the car be x km/h, and the speed of the bus be (x - 20) km/h.
We know that the distance between towns M and N is 280 km, and the time taken by the car to travel this distance is the same as the time taken by the bus plus 1 hour and 10 minutes.
Now, let's convert the time taken by the bus plus 1 hour and 10 minutes to hours. Since there are 60 minutes in an hour, 10 minutes is equal to 10/60 = 1/6 hours. So the total time taken by the bus is (x - 20) + 1/6 hours.
We can use the formula: Speed = Distance/Time to find the time taken by the car and the bus.
For the car, the formula becomes: Time taken by the car = Distance/Speed of the car.
Time taken by the car = 280/x hours.
For the bus, the formula becomes: Time taken by the bus = Distance/Speed of the bus.
Time taken by the bus = 280/(x - 20 + 1/6) hours.
Now, we can set up the equation based on the given information:
Time taken by the car = Time taken by the bus + 1 hour and 10 minutes.
280/x = 280/(x - 20 + 1/6) + 1 + 10/60 hours.
Now, we can simplify and solve this equation for x.
Multiply both sides of the equation by x(x - 20 + 1/6) to eliminate fractions:
280(x - 20 + 1/6) = 280x + x(x - 20 + 1/6) + (600/60)x(x - 20 + 1/6).
Simplifying further:
280x - 5600 + 280/6 = 280x + x^2 - 20x + x/6 - 20/6x + (x^2 - 20x + x/6 - 20/6).
Combining like terms:
280x - 5600 + 280/6 = 280x + x^2 - 20x + x/6 - 20/6x + x^2 - 20x + x/6 - 20/6.
Further simplification:
280x - 5600 + 46.67 = 280x + 2x^2 - 20x + 0.17x - 3.33x + 2x^2 - 20x + 0.17x - 3.33.
Combining like terms:
326.67 = 4x^2 - 40x.
Rearranging the equation:
4x^2 - 40x - 326.67 = 0.
Now, we have a quadratic equation that we can solve to find the value of x. We can use the quadratic formula to solve this equation. The quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / 2a.
For our equation, the coefficients are:
a = 4,
b = -40, and
c = -326.67.
By substituting these values into the quadratic formula, we can find the values of x. Please note that there will be two possible solutions for x, but one of them will be negative and can be discarded since speed cannot be negative.