Question

Автобус долає відстань 300км на 1 годину довше ніж автівка, швидкість якої на 10 км більше швидкості автобуса. Знайдіть швидкості автівки і автобуса.

Answer 1

Problem Analysis

We are given that a bus covers a distance of 300 km in 1 hour longer than a car, and the car's speed is 10 km/h faster than the bus's speed. We need to find the speeds of the car and the bus.

Solution

Let's assume the speed of the bus is x km/h. Since the car's speed is 10 km/h faster than the bus's speed, the speed of the car can be represented as x + 10 km/h.

We are given that the bus covers a distance of 300 km in 1 hour longer than the car. This means that the time taken by the bus to cover the distance is 1 hour more than the time taken by the car.

We can use the formula time = distance / speed to set up the equation.

For the bus: 300 / x = t (where t is the time taken by the bus)

For the car: 300 / (x + 10) = t - 1 (where t - 1 is the time taken by the car, as it is 1 hour less than the bus)

Now, we can solve these two equations to find the values of x and t.

Calculation

Let's solve the equations:

From the equation for the bus: 300 / x = t, we can rearrange it to find t in terms of x: t = 300 / x (Equation 1)

From the equation for the car: 300 / (x + 10) = t - 1, we can rearrange it to find t in terms of x: t = 300 / (x + 10) + 1 (Equation 2)

Now, we can equate the two expressions for t and solve for x:

300 / x = 300 / (x + 10) + 1

To simplify the equation, we can multiply both sides by x(x + 10) to eliminate the denominators:

300(x + 10) = 300x + x(x + 10)

Expanding and simplifying:

300x + 3000 = 300x + x^2 + 10x

Rearranging and simplifying:

x^2 + 10x - 3000 = 0

Now, we can solve this quadratic equation to find the value of x.

Using the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a, where a = 1, b = 10, and c = -3000, we can calculate the values of x.

Calculating the discriminant: b^2 - 4ac = 10^2 - 4(1)(-3000) = 100 + 12000 = 12100

Taking the square root of the discriminant: sqrt(12100) = 110

Using the quadratic formula:

x = (-10 ± 110) / 2(1)

Simplifying:

x = (-10 + 110) / 2 = 100 / 2 = 50 x = (-10 - 110) / 2 = -120 / 2 = -60

Since speed cannot be negative, we discard the negative value of x.

Therefore, the speed of the bus is 50 km/h.

The speed of the car can be calculated by adding 10 km/h to the speed of the bus:

Speed of car = 50 + 10 = 60 km/h

Answer

The speed of the bus is 50 km/h and the speed of the car is 60 km/h.