Question

из москвы по рижскому шоссе выехал автобус со скоростью 54км/4. через 2и1/3 вслед за ним выехал автомобиль со скоростью в 1и4/9 раза больше скорости автобуса. на каком расстоянии будет автомобиль через 1ч 45мин после своего выезда? через сколько времени после своего выезда он догонит автобус? с какой скоростью надо ехать автомобилю, чтобы догнать автобус через 3и1/2ч

Answer 1

$54\cdot1\frac49=54\cdot\frac{13}9=6\cdot13=78$ км/ч - скорость автомобиля.

1 ч 45 мин = 1,75 ч. Через это время авто проедет 78*1,75 = 136,5 км.

Пусть авто догонит автобуз через x часов. За это время авто проедет 78x км, автобус 54*(2 1/3+x), оба проедут одинаковое расстояние:

$78x = 54\cdot(2\frac13+x)\\ 78x=126+54x\\ 24x=126\\ x=5,25$

Авто догонит автобус через 5,25 ч или 5 ч 45 мин.

Чтобы догнать через 3,5 ч, авто едет со скоростью y, проезжает 3,5y км, автобус (3,5+2 1/3)*54, оба проедут одинаковое расстояние:

$3,5y = (3,5+2\frac13)\cdot54\\ 3,5y=\frac{35}6\cdot54\\ 3,5y=315\\ y=90$

Answer 2

Calculation of Distances and Time

To solve this problem, we need to calculate the distances and time traveled by both the bus and the car.

Let's start by calculating the distance traveled by the bus. We know that the bus traveled at a speed of 54 km/h for a certain amount of time. The time is given as 2 and 1/3 hours, which can also be expressed as 7/3 hours.

The distance traveled by the bus can be calculated using the formula:

Distance = Speed × Time

Substituting the values, we get:

Distance of the bus = 54 km/h × 7/3 hours

To simplify the calculation, we can convert the mixed fraction 7/3 to an improper fraction:

Distance of the bus = 54 km/h × (7/3) hours = 54 km/h × (7/3) hours = (54 × 7) km / (3 × 1) hours = 378 km / 3 hours = 126 km

So, the bus traveled a distance of 126 km.

Now, let's calculate the speed of the car. We are given that the car traveled at a speed that is 1 and 4/9 times greater than the speed of the bus.

To find the speed of the car, we can multiply the speed of the bus by 1 and 4/9:

Speed of the car = Speed of the bus × (1 + 4/9)

Substituting the value of the speed of the bus (54 km/h), we get:

Speed of the car = 54 km/h × (1 + 4/9) = 54 km/h × (13/9) = (54 × 13) km / (9 × 1) h = 702 km / 9 h = 78 km/h

So, the speed of the car is 78 km/h.

Calculation of Distance Traveled by the Car

Next, let's calculate the distance traveled by the car after 1 hour and 45 minutes (which is equivalent to 1 and 3/4 hours) from its departure.

To calculate the distance, we can use the formula:

Distance = Speed × Time

Substituting the values, we get:

Distance traveled by the car = 78 km/h × 1 and 3/4 hours

To simplify the calculation, we can convert the mixed fraction 1 and 3/4 to an improper fraction:

Distance traveled by the car = 78 km/h × (1 + 3/4) hours = 78 km/h × (7/4) hours = (78 × 7) km / (4 × 1) hours = 546 km / 4 hours = 136.5 km

So, the car will be at a distance of 136.5 km from its starting point after 1 hour and 45 minutes.

Calculation of Time Taken to Catch up to the Bus

Now, let's calculate the time it takes for the car to catch up to the bus.

To find the time, we can use the formula:

Time = Distance / Relative Speed

The relative speed is the difference in speeds between the car and the bus. In this case, the relative speed is the speed of the car minus the speed of the bus:

Relative Speed = Speed of the car - Speed of the bus

Substituting the values, we get:

Relative Speed = 78 km/h - 54 km/h = 24 km/h

Now, we can calculate the time it takes for the car to catch up to the bus:

Time taken to catch up = Distance / Relative Speed

Substituting the values, we get:

Time taken to catch up = 126 km / 24 km/h

Simplifying the calculation, we find:

Time taken to catch up = 5.25 hours

So, the car will catch up to the bus after approximately 5 hours and 15 minutes.

Calculation of Speed Required to Catch up to the Bus

Finally, let's calculate the speed at which the car needs to travel in order to catch up to the bus after 3 and 1/2 hours.

To find the required speed, we can use the formula:

Speed = Distance / Time

Substituting the values, we get:

Speed required = 126 km / 3 and 1/2 hours

To simplify the calculation, we can convert the mixed fraction 3 and 1/2 to an improper fraction:

Speed required = 126 km / (3 + 1/2) hours = 126 km / (7/2) hours = (126 × 2) km / (7 × 1) hours = 252 km / 7 hours = 36 km/h

So, the car needs to travel at a speed of 36 km/h in order to catch up to the bus after 3 and 1/2 hours.

To summarize: - The car will be at a distance of 136.5 km from its starting point after 1 hour and 45 minutes. - The car will catch up to the bus after approximately 5 hours and 15 minutes. - The car needs to travel at a speed of 36 km/h in order to catch up to the bus after 3 and 1/2 hours.