Question

Моторная лодка прошла 56км против течения реки и 32 км по течению, затратив на весь путь 3 ч. Найдите собственную скорость лодки. Если скорость реки равна 1км\ч

Answer 1

пусть х - скорость лодки 
56/(х-1)+32/(х+1)=3

Answer 2

Problem Analysis

We are given that a motorboat traveled 56 km against the current of a river and 32 km with the current, taking a total of 3 hours for the entire journey. We need to find the speed of the boat.

Solution

Let's assume the speed of the boat is x km/h and the speed of the river current is 1 km/h.

When the boat is traveling against the current, its effective speed is reduced by the speed of the current. So, the boat's speed against the current is (x - 1) km/h.

When the boat is traveling with the current, its effective speed is increased by the speed of the current. So, the boat's speed with the current is (x + 1) 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 56 km against the current is given by: 56 km = (x - 1) km/h × t1 hours The time taken to travel 32 km with the current is given by: 32 km = (x + 1) km/h × t2 hours We are also given that the total time for the journey is 3 hours: t1 + t2 = 3 hours We can solve this system of equations to find the value of x.

Solving the Equations

Let's solve equations and for t1 and t2, respectively:

From equation 56 = (x - 1) × t1 From equation 32 = (x + 1) × t2 Now, let's solve equations and simultaneously to find the values of t1 and t2.

Dividing equation by (x - 1): t1 = 56 / (x - 1)

Dividing equation by (x + 1): t2 = 32 / (x + 1)

Substituting the values of t1 and t2 into equation (56 / (x - 1)) + (32 / (x + 1)) = 3 Now, let's solve equation to find the value of x.

Calculating the Value of x

To solve equation we can multiply both sides of the equation by (x - 1)(x + 1) to eliminate the denominators:

(56(x + 1)) + (32(x - 1)) = 3(x - 1)(x + 1)

Expanding and simplifying the equation: 56x + 56 + 32x - 32 = 3(x^2 - 1)

Simplifying further: 88x + 24 = 3x^2 - 3

Rearranging the equation: 3x^2 - 88x - 27 = 0

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

Solving the Quadratic Equation

We can use the quadratic formula to solve the equation 3x^2 - 88x - 27 = 0:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 3, b = -88, and c = -27.

Substituting the values into the quadratic formula: x = (-(-88) ± √((-88)^2 - 4 * 3 * (-27))) / (2 * 3)

Simplifying: x = (88 ± √(7744 + 324)) / 6

x = (88 ± √8068) / 6

Calculating the square root of 8068: √8068 ≈ 89.92

Substituting the value of √8068 into the equation: x = (88 ± 89.92) / 6

Now, let's calculate the two possible values of x:

For x = (88 + 89.92) / 6: x ≈ 29.99

For x = (88 - 89.92) / 6: x ≈ -0.99

Since the speed of the boat cannot be negative, we can discard the solution x ≈ -0.99.

Therefore, the speed of the boat is approximately 29.99 km/h.

Answer

The speed of the boat is approximately 29.99 km/h.