Question

С лыжной базы вышла лыжница. Через 20 мин, после того как она прошла 3000 м, с этой же базы в том же направлении вышел лыжник и догнал её на расстоянии 7500 м от базы. Найдите скорость лыжницы и лыжника.

Answer 1

^ 150 м/мин; 250 м/мин. ппо

Answer 2

скорость лыжницы 150 м/мин, скорость лыжника 250 м/мин.

Answer 3

Problem Analysis

We are given the following information: - A skier leaves a ski base. - After 20 minutes, the skier has traveled 3000 meters. - Another skier leaves the same base in the same direction. - The second skier catches up to the first skier at a distance of 7500 meters from the base.

We need to find the speeds of both skiers.

Solution

Let's assume the speed of the first skier is v1 and the speed of the second skier is v2.

We can use the formula distance = speed × time to calculate the time taken by each skier.

For the first skier: - Distance = 3000 meters - Time = 20 minutes = 20/60 hours = 1/3 hours

For the second skier: - Distance = 7500 meters - Time = t hours

Since both skiers travel the same distance, we can set up the equation:

v1 × (1/3) = v2 × t

To solve for the speeds, we need another equation. We can use the fact that the second skier catches up to the first skier.

The time taken by the second skier to catch up is the same as the time taken by the first skier to travel the additional distance of 7500 - 3000 = 4500 meters.

So, we have another equation:

v2 × t = v1 × (1/3) + 4500

Now we have a system of two equations that we can solve to find the speeds of the skiers.

Let's solve the system of equations:

1. v1 × (1/3) = v2 × t 2. v2 × t = v1 × (1/3) + 4500

To eliminate the variable t, we can substitute the value of t from equation 1 into equation 2:

v2 × (v1 × (1/3)) = v1 × (1/3) + 4500

Simplifying the equation:

v2 × v1/3 = v1/3 + 4500

Multiplying both sides by 3 to eliminate the fractions:

v2 × v1 = v1 + 13500

Now we have a simpler equation:

v2 × v1 = v1 + 13500

We can rearrange the equation to solve for v2:

v2 × v1 - v1 = 13500

v1 × (v2 - 1) = 13500

v2 - 1 = 13500/v1

v2 = 13500/v1 + 1

Now we have an expression for v2 in terms of v1.

Let's substitute this expression into equation 1:

v1 × (1/3) = (13500/v1 + 1) × t

Simplifying the equation:

v1/3 = (13500/v1 + 1) × t

Multiplying both sides by 3 to eliminate the fraction:

v1 = (13500/v1 + 1) × (3t)

Since we know that t = 1/3, we can substitute this value:

v1 = (13500/v1 + 1) × (3/3)

Simplifying the equation:

v1 = 13500/v1 + 1

Multiplying both sides by v1 to eliminate the fraction:

v1^2 = 13500 + v1

Rearranging the equation:

v1^2 - v1 - 13500 = 0

Now we have a quadratic equation that we can solve to find the value of v1.

Using the quadratic formula:

v1 = (-b ± √(b^2 - 4ac)) / (2a)

where a = 1, b = -1, and c = -13500.

Solving the equation:

v1 = (-(-1) ± √((-1)^2 - 4(1)(-13500))) / (2(1))

v1 = (1 ± √(1 + 54000)) / 2

v1 = (1 ± √54101) / 2

Calculating the values of v1:

v1 ≈ 116.75 or v1 ≈ -115.75

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

Therefore, the speed of the first skier (v1) is approximately 116.75 meters per minute.

To find the speed of the second skier (v2), we can substitute the value of v1 into the expression we derived earlier:

v2 = 13500/v1 + 1

Substituting the value of v1:

v2 = 13500/116.75 + 1

Calculating the value of v2:

v2 ≈ 116.75

Therefore, the speed of the second skier (v2) is approximately 116.75 meters per minute.

Answer

The speed of the skier is approximately 116.75 meters per minute, and the speed of the second skier is also approximately 116.75 meters per minute.