Question

Расстояние между двумя пристанями равно 84 км . это расстояние катер по течению проплыл за 3ч ,а против течения за 3,5ч .найдите собственную скорость катера и скорость течения. решите системой уравнений

Answer 1

Problem Analysis

We are given the following information: - The distance between two ports is 84 km. - The boat took 3 hours to travel downstream (with the current) and 3.5 hours to travel upstream (against the current).

We need to find the speed of the boat and the speed of the current.

Let's assume the speed of the boat is represented by b and the speed of the current is represented by c.

Downstream Speed Calculation

When the boat is traveling downstream, its effective speed is the sum of its own speed and the speed of the current. Therefore, the boat's speed downstream can be calculated using the formula:

Speed downstream = Boat speed + Current speed

Upstream Speed Calculation

When the boat is traveling upstream, its effective speed is the difference between its own speed and the speed of the current. Therefore, the boat's speed upstream can be calculated using the formula:

Speed upstream = Boat speed - Current speed

Distance Calculation

The time taken to travel a certain distance is equal to the distance divided by the speed. Using this information, we can calculate the distances traveled downstream and upstream:

Distance downstream = Speed downstream * Time downstream

Distance upstream = Speed upstream * Time upstream

System of Equations

We can set up a system of equations using the given information:

Equation 1: Distance downstream = 84 km

Equation 2: Distance upstream = 84 km

Equation 3: Time downstream = 3 hours

Equation 4: Time upstream = 3.5 hours

Solution

Let's solve the system of equations to find the values of the boat's speed (b) and the current's speed (c).

From Equation 1 and Equation 3, we have:

84 = (b + c) * 3 (Equation 5)

From Equation 2 and Equation 4, we have:

84 = (b - c) * 3.5 (Equation 6)

Now we can solve this system of equations to find the values of b and c.

Solving the System of Equations

To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of elimination.

Multiplying Equation 5 by 3.5 and Equation 6 by 3, we get:

3.5 * 84 = (b + c) * 3 * 3.5

3 * 84 = (b - c) * 3 * 3

Simplifying, we have:

294 = 10.5b + 10.5c (Equation 7)

252 = 9b - 9c (Equation 8)

Now, let's subtract Equation 8 from Equation 7 to eliminate the variable c:

(10.5b + 10.5c) - (9b - 9c) = 294 - 252

Simplifying, we get:

1.5b + 19.5c = 42 (Equation 9)

Now we have a single equation with only one variable. Let's solve it.

Solving Equation 9 for b

To solve Equation 9 for b, we need to express b in terms of c.

Subtracting 19.5c from both sides of Equation 9, we get:

1.5b = 42 - 19.5c

Dividing both sides of the equation by 1.5, we have:

b = (42 - 19.5c) / 1.5

Now we have an expression for b in terms of c.

Substituting b in terms of c into Equation 8

Let's substitute the expression for b in terms of c into Equation 8 to solve for c.

252 = 9((42 - 19.5c) / 1.5) - 9c

Simplifying, we get:

252 = 6(42 - 19.5c) - 9c

Expanding and simplifying further, we have:

252 = 252 - 117c - 9c

Combining like terms, we get:

0 = -126c

Dividing both sides of the equation by -126, we find:

c = 0

Finding the Value of b

Now that we have the value of c, we can substitute it back into the expression for b:

b = (42 - 19.5c) / 1.5

Substituting c = 0, we get:

b = (42 - 19.5 * 0) / 1.5

Simplifying, we find:

b = 42 / 1.5

b = 28

Conclusion

After solving the system of equations, we find that the speed of the boat (b) is 28 km/h and the speed of the current (c) is 0 km/h.

Therefore, the boat's speed is 28 km/h and there is no current.

Please note that the solution assumes a constant speed of the boat and current throughout the journey.