Question

Катер прошёл по течению 30км, а против течения 24 за 9ч. Чему равна собственная скорость катера , если течение реки 3км/ч? 8класс. Через дискриминант надо решать, вроде.

Answer 1

Пусть х - скорость катера.
30/(х+3)+24(x-3)=9
30x-90+24x+72=9(x²-9)
54x-18=9x²-81
9x²-54x-63=0 I÷9
x²-6x-7=0 D=64
x₁=7 x₂=-1 x₂∉
Ответ: скорость катера 7 км/ч.

Answer 2

Problem Analysis

We are given that a boat traveled 30 km downstream and 24 km upstream in a total of 9 hours. The speed of the river current is given as 3 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 own speed and the speed of the current. So, the effective speed downstream is (x + 3) km/h.

When the boat is traveling upstream, its effective speed is the difference between its own speed and the speed of the current. So, the effective speed upstream is (x - 3) km/h.

We are given that the boat traveled 30 km downstream and 24 km upstream in a total of 9 hours. We can set up the following equation based on the time and distance traveled:

30 / (x + 3) + 24 / (x - 3) = 9

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

30(x - 3) + 24(x + 3) = 9(x + 3)(x - 3)

Simplifying the equation:

30x - 90 + 24x + 72 = 9(x^2 - 9)

54x - 18 = 9x^2 - 81

Rearranging the equation:

9x^2 - 54x + 63 = 0

Now, we can solve this quadratic equation using the discriminant.

Solving the Quadratic Equation

The quadratic equation we obtained is 9x^2 - 54x + 63 = 0.

The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by D = b^2 - 4ac.

In our case, a = 9, b = -54, and c = 63.

Let's calculate the discriminant:

D = (-54)^2 - 4(9)(63)

D = 2916 - 2268

D = 648

Since the discriminant is positive, we have two real and distinct solutions for x.

The solutions for x can be found using the quadratic formula:

x = (-b ± √D) / (2a)

Substituting the values of a, b, and D:

x = (-(-54) ± √648) / (2(9))

x = (54 ± √648) / 18

Simplifying further:

x = (54 ± 18√4) / 18

x = (54 ± 18 * 2) / 18

x = (54 ± 36) / 18

Now, we can calculate the two possible values of x:

x₁ = (54 + 36) / 18 = 90 / 18 = 5

x₂ = (54 - 36) / 18 = 18 / 18 = 1

Since the speed of the boat cannot be negative, the speed of the boat in still water is 5 km/h.

Answer

The speed of the boat in still water is 5 km/h.