Question

Катер прошел 5 км против течения и 14 км по течению реки затратив на это столько времени сколько ему понадобилось бы для прохождения 18 км по озеру. Какова собственая скорость катера если известно что скорость течения реки равна 3 км/ч

Answer 1

Х - собственная скорость катера
х+1 - скорость по течению
х-3 - скорость против течения
5/(х-3)+14/(х+3)=18/х  (умножим на х(х-3)(х+3))
5х(х+3)+14х(х-3)=18(х-3)(х+3)
5х2+15х+14х2-42х=18(х2+3х-3х-9)
19х2-27х=18х2-162
х2-27х+162=0
D=27*27-4*162=729-648=81   Корень из D=9
х(1)=(27-9):2=18:2=9 (км/ч)
х(2)=(27+9):2=36:2=18 (км/ч)
Ответ: собственная скорость катера равна или 9 км/ч, или 18 км/ч

Answer 2

Problem Analysis

We are given that a boat traveled 5 km against the current of a river and then 14 km with the current. We need to find the boat's own speed if the speed of the river's current is known to be 3 km/h. We can solve this problem using the concept of relative velocity.

Solution

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

When the boat is traveling against the current, 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. The distance traveled against the current is 5 km. We can use the formula distance = speed × time to find the time taken to travel this distance:

5 = (x - 3) × t1 When the boat is traveling with the current, 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. The distance traveled with the current is 14 km. We can use the same formula to find the time taken to travel this distance:

14 = (x + 3) × t2 We are also given that the time taken to travel 18 km on a lake is the same as the total time taken to travel 5 km against the current and 14 km with the current. Let's call this common time t. We can set up the equation:

18 = x × t Now we have a system of three equations with three unknowns: x, t1, and t2. We can solve this system of equations to find the boat's own speed.

Solving the System of Equations

We can solve the system of equations using substitution or elimination. Let's use substitution:

From equation we can express t in terms of x:

t = 18 / x

Substituting this value of t in equations and we get:

5 = (x - 3) × (18 / x)

14 = (x + 3) × (18 / x)

Simplifying these equations, we get:

5x = 18(x - 3)

14x = 18(x + 3)

Expanding and simplifying further, we get:

5x = 18x - 54

14x = 18x + 54

Simplifying again, we get:

13x = 54

4x = -54

Solving these equations, we find:

x = 54 / 13 ≈ 4.15

So, the boat's own speed is approximately 4.15 km/h.

Answer

The boat's own speed is approximately 4.15 km/h.