A vehicle has an engine that is connected directly to a gear box. In the highest gear, a 2 inch radius gear is turning a 14 inch diameter gear. This gear is connected to the drive shaft. The drive shaft attaches to a wheel and a tire that measures 2 and 1/3 feet in diameter. If the engine is pushing at 2500 rpm, how fast is the vehicle moving in miles per hour? Round your answer to the nearest tenth of a mile per hour.
( I seen the answer posted before but need an explanation on how you got the numbers, Thanks)
To determine how fast the vehicle is moving in miles per hour, we need to consider the gear ratios and the rotations per minute (RPM) of the engine.
First, let's calculate the gear ratio. The given gear has a 2 inch radius, so its diameter is 4 inches. Therefore, the gear connected to the driveshaft has a diameter of 14 inches. The gear ratio is determined by comparing the number of teeth on the two gears. Since we only have information about the diameters, we can use the formula:
Gear Ratio = (Diameter of Driven Gear) / (Diameter of Driving Gear)
In this case, the gear ratio is (14 inches) / (4 inches) = 3.5.
Next, we need to convert the RPM of the engine to the RPM of the driveshaft. Since the two gears are directly connected, they rotate at the same speed. Therefore, the RPM of the driveshaft is also 2500.
Now, let's calculate the linear speed of the driveshaft in feet per minute.
Circumference of the driveshaft = π × diameter of the driveshaft
= 3.14159 × (2 and 1/3 feet)
≈ 3.14159 × 2.33
≈ 7.3308 feet
Linear Speed of the driveshaft (feet per minute) = Circumference × RPM
= 7.3308 × 2500
= 18327 feet per minute
Finally, let's convert the linear speed to miles per hour.
There are 5280 feet in a mile and 60 minutes in an hour.
Speed (miles per hour) = (Linear Speed in feet per minute × 60) / (5280)
= (18327 × 60) / (5280)
≈ 208.09 mph
Therefore, the vehicle is moving approximately at a speed of 208.1 mph.
To calculate the speed of the vehicle in miles per hour, we need to find the speed of the vehicle and convert it from rpm (revolutions per minute) to mph (miles per hour).
First, let's find the speed of the gear connected to the drive shaft.
1. The gear ratio can be calculated by dividing the diameter of the larger gear by the diameter of the smaller gear:
Gear ratio = 14 inches / (2 inches * 2) = 14/4 = 3.5
2. Since the drive shaft is directly connected to the gear with the 14 inch diameter, it will rotate at the same speed:
Drive shaft speed = Engine speed = 2500 rpm
3. Calculate the rotational speed of the wheel using the gear ratio:
Wheel speed = Drive shaft speed / Gear ratio = 2500 rpm / 3.5 = 714.29 rpm
Next, let's convert the wheel speed from rpm to mph.
4. Calculate the circumference of the tire using its diameter:
Circumference = π * Diameter = π * 28/3 inches
5. Convert the rpm to revolutions per hour:
Revolutions per hour = Wheel speed * 60 minutes * 60 seconds
6. Convert the circumference of the tire from inches to miles:
1 mile = 5280 feet = 5280 * 12 inches
7. Calculate the vehicle speed in mph:
Speed (mph) = (Revolutions per hour * Circumference) / (inches in a mile)
Substituting the known values:
Speed (mph) = (Wheel speed * 60 minutes * 60 seconds * π * (28/3) inches) / (5280 * 12 inches)
Simplifying the equation, we get:
Speed (mph) ≈ (714.29 * 60 * 60 * π * (28/3)) / (5280 * 12) mph
Evaluating this expression will give us the approximate speed in miles per hour.