Question

Теплоход проходит по течению реки до пункта назначения 525 км и после стоянки возвращается в пункт назначения. Найдите скорость течения, если скорость теплохода в неподвижной воде равна 23 км/ч, стоянка длится 3 часа, а в пункт отправления теплоход возвращается через 49 часов после отправления из него. Ответ дайте в км/ч.

Answer 1

пусть скорость  течения  х км/час
скорость теплохода по течению х+23 км/час

скорость теплохода против течения  23-х км/час

49-3=46  часов  время  в пути

525: (х+23)+ 525:( 23-х)=46

525*(23-х)+525* (х+23)=46* (23²-х²)

12075-525х+525х+12075=24334-46х²

24150-24334=-46х²

46х²=184

х²=4

х=2 км /час  скорость течения,

Answer 2

Problem Analysis

We are given the following information: - The distance from the starting point to the destination is 525 km. - The speed of the boat in still water is 23 km/h. - The boat stops for 3 hours at the destination. - The boat returns to the starting point 49 hours after leaving it.

We need to find the speed of the river current.

Solution

Let's assume the speed of the river current is x km/h.

When the boat is going from the starting point to the destination, it is moving against the current. So the effective speed of the boat is the difference between its speed in still water and the speed of the current: (23 - x) km/h.

When the boat is returning from the destination to the starting point, it is moving with the current. So the effective speed of the boat is the sum of its speed in still water and the speed of the current: (23 + x) km/h.

We can use the formula speed = distance / time to calculate the time taken for each leg of the journey.

Let's calculate the time taken for the boat to reach the destination: - Distance: 525 km - Speed: (23 - x) km/h - Time: 525 / (23 - x) hours

Next, let's calculate the time taken for the boat to return to the starting point: - Distance: 525 km - Speed: (23 + x) km/h - Time: 525 / (23 + x) hours

According to the given information, the boat returns to the starting point 49 hours after leaving it. So the total time for the return journey is the time taken to reach the destination plus the 3-hour stopover plus 49 hours: - Total time: 525 / (23 - x) + 3 + 49 hours

Since the boat covers the same distance in both directions, the total time for the return journey should be equal to the time taken to reach the destination: - 525 / (23 - x) + 3 + 49 = 525 / (23 + x)

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

Calculation

Let's solve the equation to find the value of x.

525 / (23 - x) + 3 + 49 = 525 / (23 + x)

To simplify the equation, let's multiply both sides by (23 - x)(23 + x) to eliminate the denominators:

525(23 + x) + 3(23 - x)(23 + x) + 49(23 - x)(23 + x) = 525(23 - x)

Expanding and simplifying the equation:

525(23) + 525(x) + 3(23^2 - x^2) + 49(23^2 - x^2) = 525(23) - 525(x)

12075 + 525x + 3(529 - x^2) + 49(529 - x^2) = 12075 - 525x

12075 + 525x + 1587 - 3x^2 + 25821 - 49x^2 = 12075 - 525x

-52x^2 + 525x + 25821 + 1587 = 0

-52x^2 + 525x + 27408 = 0

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

Quadratic Equation Solution

The quadratic equation ax^2 + bx + c = 0 can be solved using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / (2a)

In our case, a = -52, b = 525, and c = 27408.

Substituting the values into the quadratic formula:

x = (-525 ± sqrt(525^2 - 4(-52)(27408))) / (2(-52))

Simplifying the equation:

x = (-525 ± sqrt(275625 - (-56832))) / (-104)

x = (-525 ± sqrt(332457)) / (-104)

x = (-525 ± 576.98) / (-104)

Calculating the two possible values of x:

x1 = (-525 + 576.98) / (-104) ≈ 0.52 km/h
x2 = (-525 - 576.98) / (-104) ≈ -10.27 km/h

Since the speed of the river current cannot be negative, the speed of the current is approximately 0.52 km/h.

Answer

The speed of the river current is approximately 0.52 km/h.