Question

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

Answer 1

Пусть сосбтсвенная скорость катера равна равна х км\ч, тогда скорость катера по течению реки равна х+2 км\ч, против течения реки х-2 км\час, время сплава по течению реки 40/(x+2) ч, против течения 6/(x-2). По условию задачи

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

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

40x-80+6x+12=3(x^2-4)

46x-68=3x^2-12

3x^2-46x+56=0

D=1444=38^2

x1=(46-38)/(3*2)=4/3<2 - невозможно - иначе катер не смог бы плыть против течения, он стоял бы на месте

х2=(46+38)/(3*2)=14 

x=14     ОТВЕТ : 14КМ/Ч

Answer 2

Пусть сосбтсвенная скорость катера равна равна х км\ч, тогда скорость катера по течению реки равна х+2 км\ч, против течения реки х-2 км\час, время сплава по течению реки 40/(x+2) ч, против течения 6/(x-2). По условию задачи
40/(x+2)+6/(x-2)=3
40(x-2)+6(x+2)=3(x+2)(x-2)
40x-80+6x+12=3(x^2-4)
46x-68=3x^2-12
3x^2-46x+56=0
D=1444=38^2
x1=(46-38)/(3*2)=4/3<2 - невозможно - иначе катер не смог бы плыть против течения, он стоял бы на месте
х2=(46+38)/(3*2)=14
x=14
ответ: 14 к/ч

Answer 3

Problem Analysis

We are given that a boat traveled 40 km downstream and 6 km upstream, taking a total of 3 hours for the entire journey. The speed of the current is given as 2 km/h. We need to find the boat's own speed.

Solution

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

To calculate the time taken to travel downstream, we divide the distance by the sum of the boat's speed and the current's speed: Time downstream = 40 / (x + 2) hours.

To calculate the time taken to travel upstream, we divide the distance by the difference between the boat's speed and the current's speed: Time upstream = 6 / (x - 2) hours.

Since the total time for the journey is 3 hours, we can write the equation: Time downstream + Time upstream = 3.

Substituting the values, we get: (40 / (x + 2)) + (6 / (x - 2)) = 3.

To solve this equation, we can multiply through by (x + 2)(x - 2) to eliminate the denominators: 40(x - 2) + 6(x + 2) = 3(x + 2)(x - 2).

Simplifying the equation: 40x - 80 + 6x + 12 = 3(x^2 - 4).

46x - 68 = 3x^2 - 12.

Rearranging the equation: 3x^2 - 46x + 56 = 0.

Now we can solve this quadratic equation to find the value of x.

Quadratic Equation Solution

Using the quadratic formula, we have: x = (-b ± √(b^2 - 4ac)) / (2a).

For our equation, a = 3, b = -46, and c = 56.

Substituting the values, we get: x = (-(-46) ± √((-46)^2 - 4 * 3 * 56)) / (2 * 3).

Simplifying further: x = (46 ± √(2116 - 672)) / 6.

x = (46 ± √1444) / 6.

x = (46 ± 38) / 6.

We have two possible solutions: 1. x = (46 + 38) / 6 = 84 / 6 = 14. 2. x = (46 - 38) / 6 = 8 / 6 = 4/3.

Since the speed of the boat cannot be negative, we discard the second solution.

Answer

Therefore, the boat's own speed is 14 km/h.

Verification

Let's verify our answer using the given information.

The time taken to travel downstream is 40 / (14 + 2) = 40 / 16 = 2.5 hours.

The time taken to travel upstream is 6 / (14 - 2) = 6 / 12 = 0.5 hours.

The total time for the journey is 2.5 + 0.5 = 3 hours, which matches the given information.

Hence, our answer is verified.

Conclusion

The boat's own speed is 14 km/h.