Question

Задача на движение:Теплоход прошел по течению реки 96 км и столько же против течения, затратив на весь путь 10 часов. Скорость течения 4 км/ч.Найти (в км/ч) скорость теплохода в стоячей воде.

Answer 1

Х км/ч - собственная скорость. (х+4) км/ч - скорость по течению , время затраченное на путь 96км равно 96/(х+4)ч (х-4) км/ч - скорость против течения, время затраченное на путь 96км равно 96/(х-4). Весь путь пройден за 10ч. 96/(х+4)+96/(х-4)=10 96(х-4+х+4)=10(х+4)(х-4) 96х=5х²-80 5х²-96х-80=0 D=96²+4*5*80=10816=104² х=(96+104)/10=20км/ч

Answer 2

Problem Analysis

We are given that a boat traveled 96 km downstream (with the current) and the same distance upstream (against the current), taking a total of 10 hours for the entire journey. The speed of the current is given as 4 km/h. We need to find the speed of the boat in still water.

Let's assume the speed of the boat in still water is x km/h.

Downstream Journey

When the boat is traveling downstream, it gets a boost from the current, which adds to its speed. The effective speed of the boat is the sum of its speed in still water and the speed of the current. Therefore, the effective speed of the boat during the downstream journey is (x + 4) km/h.

The distance traveled downstream is given as 96 km. We can use the formula distance = speed × time to find the time taken for the downstream journey.

Let's denote the time taken for the downstream journey as t1 hours.

Therefore, we have the equation: 96 = (x + 4) × t1.

Upstream Journey

When the boat is traveling upstream, it has to overcome the resistance of the current, which reduces its effective speed. The effective speed of the boat during the upstream journey is the difference between its speed in still water and the speed of the current. Therefore, the effective speed of the boat during the upstream journey is (x - 4) km/h.

The distance traveled upstream is also given as 96 km. We can use the formula distance = speed × time to find the time taken for the upstream journey.

Let's denote the time taken for the upstream journey as t2 hours.

Therefore, we have the equation: 96 = (x - 4) × t2.

Total Journey Time

The total time taken for the entire journey is given as 10 hours. Therefore, we have the equation: t1 + t2 = 10.

Solving the Equations

We now have a system of three equations with three unknowns: 1. 96 = (x + 4) × t1 2. 96 = (x - 4) × t2 3. t1 + t2 = 10

We can solve this system of equations to find the value of x, which represents the speed of the boat in still water.

Let's solve the equations:

From equation 1, we can express t1 in terms of x: t1 = 96 / (x + 4)

From equation 2, we can express t2 in terms of x: t2 = 96 / (x - 4)

Substituting these expressions for t1 and t2 into equation 3, we get: 96 / (x + 4) + 96 / (x - 4) = 10

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

Calculation

Let's solve the equation 96 / (x + 4) + 96 / (x - 4) = 10 to find the value of x.

Using algebraic manipulation, we can simplify the equation as follows:

96(x - 4) + 96(x + 4) = 10(x + 4)(x - 4)

Simplifying further:

96x - 384 + 96x + 384 = 10(x^2 - 16)

192x = 10x^2 - 160

10x^2 - 192x - 160 = 0

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

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 10, b = -192, and c = -160.

Substituting the values into the formula:

x = (-(-192) ± √((-192)^2 - 4 * 10 * (-160))) / (2 * 10)

Simplifying further:

x = (192 ± √(36864 + 6400)) / 20

x = (192 ± √43264) / 20

x = (192 ± 208) / 20

Now, we have two possible values for x:

1. x = (192 + 208) / 20 = 400 / 20 = 20 km/h 2. x = (192 - 208) / 20 = -16 / 20 = -0.8 km/h

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

Therefore, the speed of the boat in still water is 20 km/h.

Answer

The speed of the boat in still water is 20 km/h.