Question

На соревнованиях по картингу по кольцевой трассе один из картов проходил круг на 5 минут медленнее другого и через час отстал от него на круг. За сколько минут каждый карт проходил круг? Решение полностью.

Answer 1

Один из картов проходил круг на 5 минут медленнее (t₂= t₁+5) другого (t₁) и через час (60) отстал от него на круг (S₂= S₁-1).

V₁= 1/t₁ (1 круг за t₁ минут)
t₂= t₁+5
V₂= 1/(t₁+5)

S₂= S₁-1 (кругов)

V₂= S₂/60 <=> 1/(t₁+5) = (S₁-1)/60
S₁= V₁·60 <=> S₁= 60/t₁

1/(t₁+5) = [(60/t₁) -1]/60 <=> (60-t₁)/60t₁ - 1/(t₁+5) =0 <=>
[(60-t₁)(t₁+5) -60t₁] / 60t₁(t₁+5) =0 <=>
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60t₁ -t₁² +300 -5t₁ -60t₁ =0 <=> t₁² +5t₁ -300 =0 <=>
[ t₁= -20 (t₁>0)
[ t₁=15
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Ответ:
Один карт проходил круг за 15 мин, другой - за 20 мин.

Answer 2

1ч=60мин х-время быстрого на круг х+5-время медленного на круг 60/х- кол-во кругов быстрого 60/(х+5)- кол-во кругов медленного 60/х-60/(х+5)=1 60/х-60/(х+5)-1=0 домножим на х(х+5)=х²+5х 60(х+5)-60х-х²-5х=0 60х+300-60х-х²-5х=0 300-х²-5х=0 х²+5х-300=0 D= 52 - 4·1·(-300) = 25 + 1200 = 1225 х=-20 -не подходит х=15мин-время быстрого на круг 15+5=20мин-время медленного на круг

Answer 3

Problem Analysis

We are given that on a kart racing competition on a circular track, one kart completes a lap 5 minutes slower than another kart. Additionally, after one hour, the slower kart is one lap behind the faster kart. We need to determine the time it takes for each kart to complete one lap.

Solution

Let's assume that the faster kart takes x minutes to complete one lap. Therefore, the slower kart takes x + 5 minutes to complete one lap.

We are also given that after one hour (60 minutes), the slower kart is one lap behind the faster kart. This means that in 60 minutes, the faster kart completes 60 / x laps, while the slower kart completes 60 / (x + 5) laps.

Since the slower kart is one lap behind the faster kart, we can set up the following equation:

(60 / x) - (60 / (x + 5)) = 1

To solve this equation, we can multiply both sides by x(x + 5) to eliminate the denominators:

60(x + 5) - 60x = x(x + 5)

Simplifying the equation:

300 = x^2 + 5x

Rearranging the equation:

x^2 + 5x - 300 = 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 = 1, b = 5, and c = -300:

x = (-5 ± √(5^2 - 4(1)(-300))) / (2(1))

Simplifying further:

x = (-5 ± √(25 + 1200)) / 2

x = (-5 ± √1225) / 2

x = (-5 ± 35) / 2

We have two possible solutions for x:

1. x = (-5 + 35) / 2 = 30 / 2 = 15 2. x = (-5 - 35) / 2 = -40 / 2 = -20

Since the time to complete a lap cannot be negative, we discard the second solution.

Therefore, the faster kart takes 15 minutes to complete one lap, and the slower kart takes 15 + 5 = 20 minutes to complete one lap.

Answer

The faster kart completes one lap in 15 minutes, while the slower kart completes one lap in 20 minutes.