Question

Велосипедист проехал 67 км за 4 ч,причем на последних 27 км пути его скорость была на 2км/ч больше,чем предыдущем участке пути.Сколько времени затратил велосипедист на последние 27 км пути?

Answer 1

1.67-27=40(км)-предыдущий участок
х-скорость на предыдущем участке
х+2-скорость на участке в 27 км

$\frac{40}{x} + \frac{27}{x+2} =4 \\ \\ 40(x+2)+27x=4x(x+2) \\ 40x+80+27x=4 x^{2} +8x \\ 67x+80=4 x^{2} +8x \\ 4 x^{2} -59x-80=0 \\ D=3481+1280=4761 \\ \sqrt{4761=69} \\ \\ x1= \frac{59+69}{8} = \frac{128}{8} =16 \\ \\ x2= \frac{59-69}{8} = \frac{-10}{8} <0 \\ \\ \frac{27}{x+2} = \frac{27}{16+2} = \frac{27}{18} =1.5$

Ответ: последний участок велосипедист проехал за 1.5 часа.

Answer 2

Problem Analysis

We are given that a cyclist traveled a total distance of 67 km in 4 hours. The cyclist's speed during the last 27 km was 2 km/h faster than the speed on the previous part of the journey. We need to determine the time taken by the cyclist to cover the last 27 km.

Solution

Let's break down the problem step by step:

1. Calculate the speed of the cyclist during the first part of the journey. 2. Calculate the speed of the cyclist during the last 27 km. 3. Use the speed and distance to calculate the time taken to cover the last 27 km.

Step 1: Calculate the speed of the cyclist during the first part of the journey

To calculate the speed during the first part of the journey, we can use the formula: speed = distance / time.

Given that the total distance is 67 km and the total time is 4 hours, we can calculate the speed during the first part of the journey as follows:

speed1 = distance1 / time1

where distance1 is the distance covered during the first part of the journey and time1 is the time taken to cover that distance.

Since we don't have the specific values for distance1 and time1, we need to express them in terms of the total distance and total time.

Let's assume that the distance covered during the first part of the journey is x km. Then the distance covered during the last 27 km is (67 - x) km.

We can express time1 in terms of x and time2 (time taken to cover the last 27 km) as follows:

time1 = total time - time2

Substituting the values into the formula, we get:

speed1 = x / (total time - time2)

Step 2: Calculate the speed of the cyclist during the last 27 km

We are given that the speed during the last 27 km was 2 km/h faster than the speed during the first part of the journey. Therefore, we can express the speed during the last 27 km as:

speed2 = speed1 + 2

Step 3: Calculate the time taken to cover the last 27 km

To calculate the time taken to cover the last 27 km, we can use the formula: time = distance / speed.

Substituting the values into the formula, we get:

time2 = (67 - x) / speed2

Now we have two equations with two unknowns (x and time2). We can solve these equations simultaneously to find the values.

Let's solve the equations:

Equation 1: speed1 = x / (total time - time2)

Equation 2: time2 = (67 - x) / speed2

Substituting the value of speed1 from Equation 2 into Equation 1, we get:

x / (total time - time2) = x / (67 - x) + 2

Simplifying the equation, we get:

x(67 - x) = x(total time - time2) + 2x(67 - x)

Expanding and rearranging the equation, we get:

67x - x^2 = x(total time - time2) + 134 - 2x^2

Simplifying further, we get:

x^2 - x(total time - time2) - 67x + 134 = 0

Now we can solve this quadratic equation to find the value of x. Once we have the value of x, we can substitute it into Equation 2 to find the value of time2.

Let's solve the quadratic equation:

x^2 - x(total time - time2) - 67x + 134 = 0

Using the quadratic formula, we get:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a = 1, b = -(total time - time2 + 67), and c = 134.

Substituting the values, we get:

x = (-(total time - time2 + 67) ± √((total time - time2 + 67)^2 - 4(1)(134))) / (2(1))

After finding the value of x, we can substitute it into Equation 2 to find the value of time2.

Final Answer

To find the time taken by the cyclist to cover the last 27 km, we need to solve the quadratic equation and substitute the value of x into Equation 2.

Please note that the specific values of "total time" and "time2" are not provided in the question. Without these values, we cannot provide a numerical answer. However, the steps outlined above can be followed to find the solution once the specific values are known.