Question

Моторная лодка прошла по течению реки 16 км, а против течения 6 км, затратив на весь путь 1,5 ч. Найдите собственную скорость моторной лодки, если скорость течения реки равна 2 км/ч.

Answer 1

Х км/ч - скорость лодки
х+2 км/ч - скорость по течению
х-2 км/ч - скорость против течения
По течению - 16 км, против течения - 6 км, время - 1,5 часа.

16/(х+2) +6/(х-2)=1,5
16х-32+6х+12=1,5 х^2 - 6
1,5х^2 - 22х +14=0
D/4=121-21=100=±10^2
х1=(11-10)/1,5=0,67 - не подходит решению
х2=(11+10)/1,5=14(км/ч)-скорость лодки

Answer 2

Problem Analysis

We are given that a motorboat traveled 16 km downstream and 6 km upstream on a river, taking a total of 1.5 hours for the entire journey. The speed of the river current is given as 2 km/h. We need to find the speed of the motorboat.

Solution

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

When the motorboat is traveling downstream, its effective speed is the sum of its own speed and the speed of the river current. So, the effective speed downstream is (x + 2) km/h.

When the motorboat is traveling upstream, its effective speed is the difference between its own speed and the speed of the river current. So, the effective speed upstream is (x - 2) km/h.

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

The time taken to travel downstream is given by: 16 km = (x + 2) km/h × t1

The time taken to travel upstream is given by: 6 km = (x - 2) km/h × t2

We are also given that the total time for the journey is 1.5 hours: t1 + t2 = 1.5 hours

We can solve these equations to find the value of x, which represents the speed of the motorboat.

Calculation

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

From the equation 16 km = (x + 2) km/h × t1, we can rearrange it to find t1: t1 = 16 km / (x + 2) km/h

From the equation 6 km = (x - 2) km/h × t2, we can rearrange it to find t2: t2 = 6 km / (x - 2) km/h

Substituting the values of t1 and t2 into the equation t1 + t2 = 1.5 hours, we get: 16 km / (x + 2) km/h + 6 km / (x - 2) km/h = 1.5 hours

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

Solution

To solve the equation, we can multiply both sides by the least common multiple (LCM) of the denominators to eliminate the fractions.

Multiplying both sides by (x + 2)(x - 2), we get: 16(x - 2) + 6(x + 2) = 1.5(x + 2)(x - 2)

Expanding and simplifying the equation, we get: 16x - 32 + 6x + 12 = 1.5(x^2 - 4)

Simplifying further, we get: 22x - 20 = 1.5x^2 - 6

Rearranging the equation, we get a quadratic equation: 1.5x^2 - 22x + 26 = 0

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

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

Substituting the values of a, b, and c into the formula, we get: x = (-(-22) ± √((-22)^2 - 4(1.5)(26))) / (2(1.5))

Simplifying further, we get: x = (22 ± √(484 - 156)) / 3

Calculating the values inside the square root, we get: x = (22 ± √328) / 3

Simplifying the square root, we get: x = (22 ± √(4 × 82)) / 3

Further simplifying, we get: x = (22 ± 2√82) / 3

So, the two possible values of x are: x = (22 + 2√82) / 3 and x = (22 - 2√82) / 3

Since the speed of the motorboat cannot be negative, we can discard the negative value.

Therefore, the speed of the motorboat is approximately (22 + 2√82) / 3 km/h.

Answer

The speed of the motorboat is approximately (22 + 2√82) / 3 km/h.