Question

Катер прошёл 40 км по течению реки и 6 км против течения, затратив на весь путь 3 ч. Какова собственная скорость катера, если скорость течения 2 км/ч? решите пожалуйста))

Answer 1

Х-скорость катера

 

40:(Х+2)+6:(Х-2)=3

40(Х-2)+6(Х+2)=3(Х+2)(Х-2)

40Х-80+6Х+12=3Х^2-3*4

3х^2-46Х+56=0

D=2116-672

D=1444

корень из дискриминанта=38

 

Х1 - не годится

X2=(46+38):6=14

                                     Ответ: скорость катера 14 км/x

Answer 2

Problem Analysis

We are given that a boat traveled 40 km downstream and 6 km upstream in a total of 3 hours. The speed of the current is given as 2 km/h. We need to find the speed of the boat in still water.

Solution

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

When the boat is traveling downstream, its effective speed is the sum of its speed in still water and the speed of the current. Therefore, the boat's effective speed downstream is (x + 2) km/h.

When the boat is traveling upstream, its effective speed is the difference between its speed in still water and the speed of the current. Therefore, the boat's effective speed upstream is (x - 2) 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 40 km is given by: 40 = (x + 2) × t1 (Equation 1)

The time taken to travel upstream for 6 km is given by: 6 = (x - 2) × t2 (Equation 2)

We are also given that the total time for the journey is 3 hours: t1 + t2 = 3 (Equation 3)

We can solve this system of equations to find the value of x, which represents the speed of the boat in still water.

Solution Steps

1. Solve Equation 1 for t1 in terms of x. 2. Solve Equation 2 for t2 in terms of x. 3. Substitute the values of t1 and t2 from steps 1 and 2 into Equation 3. 4. Solve the resulting equation for x.

Let's solve the problem step by step.

Step 1: Solve Equation 1 for t1

From Equation 1, we have: 40 = (x + 2) × t1

Solving for t1, we get: t1 = 40 / (x + 2) (Equation 4)

Step 2: Solve Equation 2 for t2

From Equation 2, we have: 6 = (x - 2) × t2

Solving for t2, we get: t2 = 6 / (x - 2) (Equation 5)

Step 3: Substitute t1 and t2 into Equation 3

From Equation 3, we have: t1 + t2 = 3

Substituting the values of t1 and t2 from steps 1 and 2, we get: 40 / (x + 2) + 6 / (x - 2) = 3 (Equation 6)

Step 4: Solve the resulting equation for x

To solve Equation 6 for x, we can multiply both sides of the equation by (x + 2)(x - 2) to eliminate the denominators:

40(x - 2) + 6(x + 2) = 3(x + 2)(x - 2)

Expanding and simplifying the equation, we get: 40x - 80 + 6x + 12 = 3(x^2 - 4)

Simplifying further, we have: 46x - 68 = 3x^2 - 12

Rearranging the equation, we get a quadratic equation: 3x^2 - 46x + 56 = 0

We can solve this quadratic equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, the values of a, b, and c are: a = 3 b = -46 c = 56

Solving the quadratic equation, we find two possible values for x.

Let's calculate the values of x.

Calculation

Using the quadratic formula, we have: x = (-(-46) ± √((-46)^2 - 4 * 3 * 56)) / (2 * 3)

Simplifying the equation, we get: x = (46 ± √(2116 - 672)) / 6

x = (46 ± √1444) / 6

x = (46 ± 38) / 6

Therefore, the two possible values for x are: x1 = (46 + 38) / 6 = 84 / 6 = 14 x2 = (46 - 38) / 6 = 8 / 6 = 4/3

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

Answer

The speed of the boat in still water is 14 km/h.

Please let me know if you need any further assistance.