Question

Баржа шла по течению реки 64 км и, повернула обратно, прошла ещё 48 км, затратив на весь путь 8 часов. Найдите собственную скорость баржи, если скорость течения реки равна 5км/ч.

Answer 1

$\frac{64}{v+5}+ \frac{48}{v-5}=8 \\ 64(v-5)+48(v+5)=8(v-5)(v+5) \\ 64v-320 +48v+240=8v^2-200 \\ -8v^2+112v+120=0 \\ D=b^2-4ac=16384 \\ v_1_,_2= \frac{-b^+_- \sqrt{D} }{2a} \\ v_1=15 \\ v_2=-1$
но скорость баржи не может быть отрицательной, следовательно V=15 км/ч

Answer 2

Problem Analysis

We are given that a barge travels downstream on a river for 64 km, then turns back and travels upstream for an additional 48 km. The total time taken for the entire journey is 8 hours. We are also given that the speed of the river's current is 5 km/h. We need to find the speed of the barge.

Solution

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

When the barge is traveling downstream, its effective speed is the sum of its own speed and the speed of the river's current. Therefore, the effective speed is (x + 5) km/h.

When the barge is traveling upstream, its effective speed is the difference between its own speed and the speed of the river's current. Therefore, the effective speed is (x - 5) km/h.

We can now calculate the time taken for each leg of the journey using the formula:

Time = Distance / Speed

For the downstream journey: Time taken = Distance / Effective speed = 64 km / (x + 5) km/h

For the upstream journey: Time taken = Distance / Effective speed = 48 km / (x - 5) km/h

The total time taken for the entire journey is given as 8 hours. Therefore, we can write the equation:

64 / (x + 5) + 48 / (x - 5) = 8

Now, let's solve this equation to find the value of x, which represents the speed of the barge.

Calculation

To solve the equation, we can start by simplifying it:

64 / (x + 5) + 48 / (x - 5) = 8

Multiplying both sides of the equation by (x + 5)(x - 5) to eliminate the denominators:

64(x - 5) + 48(x + 5) = 8(x + 5)(x - 5)

Expanding and simplifying:

64x - 320 + 48x + 240 = 8(x^2 - 25)

112x - 80 = 8x^2 - 200

Rearranging the equation:

8x^2 - 112x + 120 = 0

Dividing the entire equation by 8:

x^2 - 14x + 15 = 0

Now, we can solve this quadratic equation using factoring or the quadratic formula. Let's use factoring:

(x - 1)(x - 15) = 0

This gives us two possible values for x: x = 1 or x = 15.

However, we need to consider the physical context of the problem. The speed of the barge cannot be negative, so we can discard the solution x = 1.

Therefore, the speed of the barge is 15 km/h.

Answer

The speed of the barge is 15 km/h.