Question

ПОМОГИТЕ!! Катер прошел 12 км против течения реки и 5 км по течению. При этом он затратил столько времени, сколько ему потребовалось бы, если бы он шел 18 км по озеру. Какова собственная скорость катера, если известно, что скорость течения реки равна 3 км/ч? РЕШАТЬ ЧЕРЕЗ ДИСКРИМИНАНТ

Answer 1

Ответ:

Обозначим скорость катера за х, тогда скорость против течения х-3, а по течению х+3.

12/х-3 время, хатраченное против течения, а 5/х+3 время по течению. Составляем уравнение:

12/х-3+5/х+3=18/х

12(х^2+3x)+5(x^2-3x)=18(x^2-9)

12x^2+36x+5x^2-15x=18x^2-162

-x^2+21x+162=0

x^2-21x-162=0

D=441+648

D=1089

x1=(21-V1089)/2=-6-посторонний корень, т.к. скорость не может быть

x2=(21+V1089)=54/2=27

27 км/ч-собственная скорость катера

Answer 2

Problem Analysis

We are given that a boat traveled 12 km against the current of a river and then 5 km with the current. The boat took the same amount of time as it would have taken to travel 18 km on a lake. We need to find the boat's own speed, given that the speed of the river current is 3 km/h.

Solution

Let's assume the boat's own speed is x 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 effective speed is (x - 3) 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 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.

For the first leg (against the current), the time taken is given by 12 / (x - 3) hours.

For the second leg (with the current), the time taken is given by 5 / (x + 3) hours.

Since the boat took the same amount of time to travel 18 km on a lake, we can set up the equation:

12 / (x - 3) + 5 / (x + 3) = 18 / x

To solve this equation, we can multiply through by x(x - 3)(x + 3) to eliminate the denominators:

12x(x + 3) + 5x(x - 3) = 18(x - 3)(x + 3)

Simplifying the equation will give us a quadratic equation, which we can solve using the discriminant.

Calculation

Let's calculate the solution using the given equation.

12x(x + 3) + 5x(x - 3) = 18(x - 3)(x + 3)

Expanding and simplifying the equation:

12x^2 + 36x + 5x^2 - 15x = 18(x^2 - 9)

17x^2 + 21x = 18x^2 - 162

0 = x^2 - 21x - 162

Now we have a quadratic equation in the form ax^2 + bx + c = 0.

Comparing the equation with the standard form, we have:

a = 1, b = -21, c = -162

We can use the discriminant formula to find the solutions:

Discriminant (D) = b^2 - 4ac

D = (-21)^2 - 4(1)(-162)

D = 441 + 648

D = 1089

Since the discriminant is positive, we have two real solutions.

Using the quadratic formula:

x = (-b ± √D) / (2a)

x = (-(-21) ± √1089) / (2 * 1)

x = (21 ± 33) / 2

So, the two possible values for the boat's own speed are:

x1 = (21 + 33) / 2 = 54 / 2 = 27 km/h

x2 = (21 - 33) / 2 = -12 / 2 = -6 km/h

Since the speed cannot be negative, the boat's own speed is 27 km/h.

Answer

The boat's own speed is 27 km/h.

Note: The negative solution (-6 km/h) is not valid in this context, as it represents a speed in the opposite direction of the boat's motion.