Question

♡2. Катер пройшов 48 км за течією річки і повернувся назад, витративши на шлях проти течії на 3 год більше, ніж на шлях за течією річки. Знайдіть власну швидкість катера, якщо швидкість течії становить 4 КМ/ГОД.​

Answer 1

Пояснення:

Нехай власна швидкість катера дорівнює х км/год.       ⇒

$\frac{48}{x-4} -\frac{48}{x+4} =3\\48*(x+4)-48*(x-4)=3*(x-4)*(x+4)\\48x+192-48x+192=3*(x^2-16)\\384=3x^2-48\\3x^2=432\ |:3\\x^2=144\\x_1=-12\notin\ \ \ \ x_2=12\in.$

Відповідь: власна швидкість катера дорівнює 12 км/год.

Answer 2

Problem Analysis

We are given that a boat traveled 48 km downstream and then returned upstream, taking 3 hours longer for the upstream journey than the downstream journey. We need to find the speed of the boat given that the speed of the current is 4 km/h.

Solution

Let's assume the speed of the boat in still water is x km/h. Since the boat is traveling downstream, its effective speed will be the sum of its own speed and the speed of the current. Therefore, the effective speed downstream will be (x + 4) km/h.

Similarly, when the boat is traveling upstream, its effective speed will be the difference between its own speed and the speed of the current. Therefore, the effective speed upstream will be (x - 4) km/h.

We are given that the boat took 3 hours longer to travel upstream than downstream. Let's denote the time taken to travel downstream as t hours. Therefore, the time taken to travel upstream will be (t + 3) hours.

We can use the formula distance = speed × time to set up two equations based on the given information:

1. For the downstream journey: 48 = (x + 4) × t 2. For the upstream journey: 48 = (x - 4) × (t + 3)

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

Solving the Equations

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

From equation 1, we have: 48 = (x + 4) × t

From equation 2, we have: 48 = (x - 4) × (t + 3)

Expanding equation 2, we get: 48 = xt - 4t + 3x - 12

Rearranging the terms, we have: xt - 4t + 3x = 60 (equation 3)

Now, we can solve equations 1 and 3 simultaneously to find the value of x.

From equation 1, we have: 48 = (x + 4) × t

Rearranging the terms, we get: xt + 4t = 48 (equation 4)

Now, we can solve equations 3 and 4 simultaneously.

Subtracting equation 4 from equation 3, we get: (xt - 4t + 3x) - (xt + 4t) = 60 - 48

Simplifying the equation, we have: -8t + 3x - 4t = 12

Combining like terms, we get: -12t + 3x = 12 (equation 5)

Now, we have a system of two equations: xt + 4t = 48 (equation 4) -12t + 3x = 12 (equation 5)

We can solve this system of equations using any method, such as substitution or elimination. Let's use the elimination method.

Multiplying equation 4 by 3 and equation 5 by 4, we get: 3xt + 12t = 144 (equation 6) -48t + 12x = 48 (equation 7)

Adding equation 6 and equation 7, we get: (3xt + 12t) + (-48t + 12x) = 144 + 48

Simplifying the equation, we have: 3xt - 36t + 12x = 192

Rearranging the terms, we get: 3xt + 12x - 36t = 192 (equation 8)

Now, we have a system of two equations: 3xt + 12x - 36t = 192 (equation 8) -12t + 3x = 12 (equation 5)

Multiplying equation 5 by 4, we get: -48t + 12x = 48 (equation 9)

Adding equation 8 and equation 9, we get: (3xt + 12x - 36t) + (-48t + 12x) = 192 + 48

Simplifying the equation, we have: 3xt - 84t = 240

Rearranging the terms, we get: 3xt - 84t = 240 (equation 10)

Now, we have a single equation: 3xt - 84t = 240 (equation 10)

We can solve equation 10 for the value of x.

Dividing both sides of equation 10 by 3t, we get: x - 28 = 80/t

Adding 28 to both sides of the equation, we get: x = 28 + 80/t

Since we know that the speed of the current is 4 km/h, we can substitute t = 48/(x + 4) into the equation to solve for x.

Substituting the value of t into the equation, we have: x = 28 + 80/(48/(x + 4))

Simplifying the equation, we get: x = 28 + (80(x + 4))/48

Multiplying both sides of the equation by 48, we get: 48x = 48(28) + 80(x + 4)

Simplifying the equation, we have: 48x = 1344 + 80x + 320

Combining like terms, we get: 48x - 80x = 1344 + 320

Simplifying the equation, we have: -32x = 1664

Dividing both sides of the equation by -32, we get: x = -52

Since the speed of the boat cannot be negative, we can conclude that there is no solution to this problem.

Therefore, there is no valid speed for the boat that satisfies the given conditions.

Conclusion

Based on the given information, there is no valid speed for the boat that satisfies the conditions of the problem.