1.A motorcycle has a reserved fuel of 0.5 liter which can be used if it's 3-liter fuel tank is about to be emptied. The motorcycle consumes at most 0.5 liter fuel for every 20 km of travel.
a. What mathematical statement represents the amount of fuel that would be left in the motorcycle's fuel tank after traveling a certain distance if it's tank is full at the start of travel?
b. Suppose the motorcycle's tank is full and it travels a distance of 55 km, about how much fuel would be left in the tank?
c. If the motorcycle travels a distance of 130 km with its tank full, would the amount of fuel in its tank be enough to cover the given distance? Explain why?
2. A bus and a car left a place at the same time traveling in opposite direction. After 2 hours, the distance between them is at most 350 km.
a. What mathematical statement represents the distance between the two vehicles after 2 hours? Define the variables used.
b. What could be the average speed of each vehicle in kilometers per hour?
c. If the car travels at a speed of 70 kilometers per hour, what could be the maximum speed of the bus?
d. If the bus travels at a speed of 70 kilometers per hour, is it possible that the car's speed is 60 kilometers per hour? Explain or justify your answer.
e. If the car's speed is 65 kilometers per hour, is it possible that the bus speed is 75 kilometers per hour? Explain or justify your answer.
1a. Fr = 3 - 0.5L/20km * d.
Fr = 3 - 0.025d.
Fr = Fuel remaining in liters.
d = Distance traveled in km.
b. Fr = 3 - 0.025*55 = 1.625 Liters.
c. No. When d equals 130 km, Fr is negative which means the tank empties before 130 km is reached.
The max distance can be calculated by setting Fr to zero and solving for d:
Fr = 3 - 0.025d = 0.
0.025d = 3.
d = 120 km, max.
2a. D = Dc - Db.
Dc: East.
Db:West(negative).
b. V = 0.5*350km/2h = 87.5 km/h(Assuming Dc = Db).
c. D = Vc*t - Vb*t = 350 km.
70*2 - Vb*2 = 350.
-2Vb = 350-140 = 210.
Vb = -105 km/h = 105 km/h, West.
d. D = Vc*2 - (-70*2) = 350.
2Vc + 140 = 350.
2Vc = 210.
Vc = 105 km/h, East.
tabular method
a. The mathematical statement representing the amount of fuel left in the motorcycle's fuel tank after traveling a certain distance if its tank is full at the start of travel can be represented as:
Fuel left (in liters) = 3 + (Distance traveled / 20) - 0.5
b. If the motorcycle's tank is full and it travels a distance of 55 km, the amount of fuel left in the tank can be calculated:
Fuel left (in liters) = 3 + (55 / 20) - 0.5 = 3 + 2.75 - 0.5 = 5.25 - 0.5 = 4.75 liters
c. If the motorcycle travels a distance of 130 km with its tank full, the amount of fuel in its tank would not be enough to cover the given distance. This is because the motorcycle consumes at most 0.5 liters of fuel for every 20 km traveled. So, for 130 km, it would consume 0.5 liters x (130 / 20) = 3.25 liters, which exceeds the reserved fuel of 0.5 liters. Therefore, the amount of fuel in its tank would not be enough.
2. a. The mathematical statement representing the distance between the bus and the car after 2 hours can be represented as:
Distance = (Bus speed * time) + (Car speed * time)
Let's define the variables:
Bus speed = b km/h
Car speed = c km/h
Time = 2 hours (since they left at the same time)
Distance = (b * 2) + (c * 2)
b. The average speed of each vehicle in kilometers per hour can be calculated by dividing the distance traveled by the time taken:
Average speed of the bus = Distance / Time = (b * 2) / 2 = b km/h
Average speed of the car = Distance / Time = (c * 2) / 2 = c km/h
c. If the car travels at a speed of 70 kilometers per hour, the maximum speed of the bus can be calculated using the distance between them:
Distance = (bus speed * time) + (car speed * time)
350 = (bus speed * 2) + (70 * 2)
350 = 2b + 140
2b = 350 - 140
2b = 210
b = 105
The maximum speed of the bus would be 105 km/h.
d. If the bus travels at a speed of 70 kilometers per hour, it is possible that the car's speed is 60 kilometers per hour. The distance between the two vehicles would be the same for both options (70 km/h and 60 km/h), and it is not specified that the speeds have to be different.
e. If the car's speed is 65 kilometers per hour, it is not possible for the bus speed to be 75 kilometers per hour. The sum of the distances traveled by the car and the bus is fixed at 350 km after 2 hours. If the car travels at 65 km/h, it would cover a distance of 65 km/h * 2 h = 130 km. Therefore, the bus would have covered 350 km - 130 km = 220 km. If we assume the bus speed is 75 km/h, the time it would take to cover 220 km would be 220 km / 75 km/h = 2.93 hours, which is not possible within the given time frame of 2 hours.
a. The mathematical statement that represents the amount of fuel left in the motorcycle's fuel tank after traveling a certain distance if it's tank is full at the start of travel can be expressed as:
Fuel left (in liters) = 3 - (distance traveled / 20)
b. If the motorcycle's tank is full and it travels a distance of 55 km, we can calculate the amount of fuel left in the tank using the formula from part a:
Fuel left = 3 - (55 / 20)
Fuel left = 3 - 2.75
Fuel left = 0.25 liters
Therefore, about 0.25 liters of fuel would be left in the tank.
c. If the motorcycle travels a distance of 130 km with its tank full, we can again use the formula from part a to determine the amount of fuel left in the tank:
Fuel left = 3 - (130 / 20)
Fuel left = 3 - 6.5
Fuel left = -3.5 liters
Since the fuel left is negative (-3.5 liters), it means that the amount of fuel in the tank would not be enough to cover the given distance of 130 km. The motorcycle would run out of fuel before completing the journey.
a. The mathematical statement that represents the amount of fuel left in the motorcycle's fuel tank after traveling a certain distance when the tank is full at the start of travel can be represented as follows:
Fuel left in tank (in liters) = 3 - (distance traveled / 20)
b. If the motorcycle's tank is full and it travels a distance of 55 km, we can substitute the value of distance traveled (55) into the equation from part a to find out how much fuel would be left in the tank:
Fuel left in tank (in liters) = 3 - (55 / 20)
Fuel left in tank (in liters) = 3 - 2.75
Fuel left in tank (in liters) = 0.25
Therefore, about 0.25 liters of fuel would be left in the tank.
c. If the motorcycle travels a distance of 130 km with its tank full, we can once again substitute the value of distance traveled (130) into the equation from part a to determine if the amount of fuel in the tank would be enough to cover the given distance:
Fuel left in tank (in liters) = 3 - (130 / 20)
Fuel left in tank (in liters) = 3 - 6.5
Fuel left in tank (in liters) = -3.5
The result is a negative value, which implies that the fuel in the tank would not be enough to cover the given distance of 130 km. Thus, the amount of fuel in the tank would not be sufficient.
2. a. The mathematical statement that represents the distance between the two vehicles after 2 hours can be represented as:
Distance between bus and car (in kilometers) = (speed of bus * 2) + (speed of car * 2)
Let's define the variables used:
- Distance between bus and car: the distance traveled by the bus and the car after 2 hours
- Speed of bus: the speed at which the bus is traveling in kilometers per hour
- Speed of car: the speed at which the car is traveling in kilometers per hour
b. To find the average speed of each vehicle in kilometers per hour, we need to divide the distance traveled by the time taken. Since both vehicles have traveled for 2 hours, we can use the formula:
Average speed of vehicle = Distance traveled / Time taken
c. If the car travels at a speed of 70 kilometers per hour, we can substitute the speed of car (70) into the equation from part b to find the maximum speed of the bus:
Distance between bus and car (in kilometers) = (speed of bus * 2) + (70 * 2)
350 = (speed of bus * 2) + 140
(speed of bus * 2) = 350 - 140
(speed of bus * 2) = 210
speed of bus = 210 / 2
speed of bus = 105 kilometers per hour
Therefore, the maximum speed of the bus would be 105 kilometers per hour.
d. If the bus travels at a speed of 70 kilometers per hour, it is not possible for the car's speed to be 60 kilometers per hour. This is because the distance between the two vehicles after 2 hours is at most 350 km, and the sum of their speeds after 2 hours would be:
Distance between bus and car = (70 * 2) + (60 * 2)
Distance between bus and car = 140 + 120
Distance between bus and car = 260 kilometers
Since the actual distance between the vehicles after 2 hours is at most 350 km, it is not possible for the car's speed to be 60 kilometers per hour.
e. If the car's speed is 65 kilometers per hour, we can once again substitute the speed of car (65) into the equation from part c to find out if the bus speed can be 75 kilometers per hour:
Distance between bus and car (in kilometers) = (speed of bus * 2) + (65 * 2)
350 = (speed of bus * 2) + 130
(speed of bus * 2) = 350 - 130
(speed of bus * 2) = 220
speed of bus = 220 / 2
speed of bus = 110 kilometers per hour
Therefore, the maximum speed of the bus would be 110 kilometers per hour. Hence, it is not possible for the bus speed to be 75 kilometers per hour if the car's speed is 65 kilometers per hour.