Question

Расстояние между двумя городами 93 км. Из одного города в другой выехал велосипедист. Через час навстречу ему из второго города выехал второй велосипедист, скорость которого была на 3 км/ч больше скорости первого. Велосипедисты встретились на расстоянии 45 км от первого города. Найдите скорость каждого велосипедиста

Answer 1

Пусть х - скорость первого велосипедиста, тогда скорость второго велосипедиста х+3. До места встречи первый велосипедист затратил 45/х часов, а второй (93-45=48км) 48/(х+3) часа, но так как второй велосипедист выехал на 1 час позже, то уравнение будет выглядеть так:
45/х=48/(x+3)+1
45/x-48/(x+3)=1
45(x+3)-48x=x²+3x
45x+135-48x-x²-3x=0
-x²-6x+135=0
D=36+540=576
x₁=(6-24)/(-2)=9     x₂=(6+24)/(-2)=-15
Скорость первого велосипедиста 9 км/ч, а второго 9+3=12 км/ч.

Answer 2

Problem Analysis

We are given that the distance between two cities is 93 km. A cyclist starts from one city and after an hour, another cyclist starts from the second city. The second cyclist's speed is 3 km/h faster than the first cyclist. They meet each other at a distance of 45 km from the first city. We need to find the speed of each cyclist.

Solution

Let's assume the speed of the first cyclist is x km/h. Since the second cyclist's speed is 3 km/h faster, the speed of the second cyclist is x + 3 km/h.

We can use the formula: Speed = Distance / Time to find the time taken by each cyclist to meet each other.

The first cyclist travels for 1 hour before meeting the second cyclist. So, the time taken by the first cyclist to meet the second cyclist is 1 + t hours, where t is the time taken by the second cyclist to meet the first cyclist.

Using the formula, we can write the equation: 45 / (1 + t) = x (equation 1) 45 / t = x + 3 (equation 2)

We can solve these two equations to find the values of x and t.

Solution Steps

1. Substitute the value of x from equation 1 into equation 2: 45 / t = 45 / (1 + t) + 3 2. Multiply both sides of the equation by t(1 + t) to eliminate the denominators: 45t(1 + t) = 45(1 + t) + 3t(1 + t) 3. Simplify the equation: 45t^2 + 45t = 45 + 45t + 3t + 3t^2 4. Rearrange the equation to form a quadratic equation: 42t^2 - 3t - 45 = 0 5. Solve the quadratic equation using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a) where a = 42, b = -3, and c = -45. 6. Calculate the values of t using the quadratic formula. 7. Substitute the values of t into equation 1 to find the value of x. 8. Calculate the value of x + 3 to find the speed of the second cyclist.

Let's calculate the values step by step.

Calculation

1. Substitute the value of x from equation 1 into equation 2: 45 / t = 45 / (1 + t) + 3

2. Multiply both sides of the equation by t(1 + t) to eliminate the denominators: 45t(1 + t) = 45(1 + t) + 3t(1 + t)

3. Simplify the equation: 45t^2 + 45t = 45 + 45t + 3t + 3t^2

4. Rearrange the equation to form a quadratic equation: 42t^2 - 3t - 45 = 0

5. Solve the quadratic equation using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a) where a = 42, b = -3, and c = -45.

Substituting the values: t = (-(-3) ± √((-3)^2 - 4 * 42 * (-45))) / (2 * 42) t = (3 ± √(9 + 10080)) / 84 t = (3 ± √10089) / 84

Taking the positive root: t = (3 + √10089) / 84

Calculating the value of t: t ≈ 0.464

6. Substitute the value of t into equation 1 to find the value of x: 45 / (1 + t) = x

Substituting the value of t: 45 / (1 + 0.464) = x

Calculating the value of x: x ≈ 30.464

7. Calculate the value of x + 3 to find the speed of the second cyclist: x + 3 ≈ 30.464 + 3 ≈ 33.464

Answer

The speed of the first cyclist is approximately 30.464 km/h, and the speed of the second cyclist is approximately 33.464 km/h.

Please note that these values are approximate due to rounding.

Let me know if you need any further assistance!