Question

лодка прошла 10 км по течению реки,затем 2 км против течения ,затратив на весь путь 1,5 часа .Найдите собственную скорость лодки,если скорость течения реки равна 3 км\ч

Answer 1

Х км/ч - собственная скорость лодки

10/(х + 3) + 2/(х - 3) = 1,5
10х - 30 + 2х + 6 - 1,5х² + 13,5 = 0
-1,5х² + 12х - 10,5 = 0
х² - 8х + 7 = 0
По теореме Виета: х₁ = 1   х₂ = 7
х₁ = 1 - не удовлетворяет условию задачи, т.к. скорость лодки не может быть меньше скорости реки

7 км/ч - собственная скорость лодки

Answer 2

Problem Analysis

We are given the following information: - The boat traveled 10 km downstream (with the current). - The boat then traveled 2 km upstream (against the current). - The total time for the entire journey was 1.5 hours. - The speed of the river current is 3 km/h.

We need to find the speed of the boat.

Solution

Let's assume the speed of the boat is x km/h.

When the boat is traveling downstream, its effective speed is the sum of its own speed and the speed of the current. So, the effective speed is (x + 3) km/h.

When the boat is traveling upstream, its effective speed is the difference between its own speed and the speed of the current. So, the effective speed is (x - 3) km/h.

We can use the formula distance = speed × time to calculate the time taken for each leg of the journey.

The time taken to travel downstream for 10 km is 10 / (x + 3) hours.

The time taken to travel upstream for 2 km is 2 / (x - 3) hours.

The total time for the entire journey is given as 1.5 hours.

So, we can set up the equation:

10 / (x + 3) + 2 / (x - 3) = 1.5

Now, let's solve this equation to find the value of x.

Calculation

To solve the equation, we can multiply both sides by the least common multiple (LCM) of the denominators to eliminate the fractions.

The LCM of (x + 3) and (x - 3) is (x + 3)(x - 3).

Multiplying both sides of the equation by (x + 3)(x - 3), we get:

10(x - 3) + 2(x + 3) = 1.5(x + 3)(x - 3)

Expanding and simplifying the equation:

10x - 30 + 2x + 6 = 1.5(x^2 - 9)

12x - 24 = 1.5x^2 - 13.5

Rearranging the equation:

1.5x^2 - 12x + 13.5 = 0

Now, we can solve this quadratic equation using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 1.5, b = -12, and c = 13.5.

Substituting the values into the formula:

x = (-(-12) ± √((-12)^2 - 4 * 1.5 * 13.5)) / (2 * 1.5)

Simplifying the equation:

x = (12 ± √(144 - 81)) / 3

x = (12 ± √63) / 3

Calculating the square root of 63:

√63 ≈ 7.937

Substituting the value into the equation:

x = (12 ± 7.937) / 3

Now, we can calculate the two possible values of x:

x₁ = (12 + 7.937) / 3 ≈ 6.312

x₂ = (12 - 7.937) / 3 ≈ 1.021

Since the speed of the boat cannot be negative, the speed of the boat is approximately 6.312 km/h.

Answer

The speed of the boat is approximately 6.312 km/h.