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.
(hint: watch your measurements)
To determine how fast the vehicle is moving in miles per hour, we need to find the rotational speed of the wheel and tire assembly and then convert it to linear speed.
First, let's find the rotational speed of the wheel and tire assembly:
1. Convert the engine speed from rpm to revolutions per second (rps):
- Divide the engine speed by 60 to convert rpm to rps.
- 2500 rpm ÷ 60 = 41.67 rps (rounded to two decimal places).
2. Determine the rotational speed of the gear connected to the drive shaft:
- Since the highest gear has a 2-inch radius, its circumference (distance traveled in one revolution) is 2π(2) = 4π inches.
- Convert the gear's rotational speed from rps to revolutions per minute (rpm):
Multiply the rotational speed by 60 to convert rps to rpm.
- 41.67 rps × 60 = 2500.2 rpm (rounded to one decimal place).
3. Calculate the rotational speed of the gear connected to the wheel and tire assembly:
- Since the gear connected to the drive shaft has a 14-inch diameter, its circumference is 14π inches.
- Divide the rotational speed of the previous gear by the circumference of this gear to find the rotational speed of the wheel and tire assembly.
- 2500.2 rpm ÷ (14π inches) = 56.651 rpm (rounded to three decimal places).
Next, let's convert the rotational speed of the wheel and tire assembly to linear speed:
4. Convert the rotational speed from rpm to miles per hour (mph):
- Multiply the rotational speed by the circumference of the wheel and tire assembly.
- The diameter of the wheel and tire assembly is 2 and 1/3 feet, so its circumference is 2π(2 and 1/3) = 4π(7/3) feet.
- Convert the circumference from feet per minute to miles per hour by multiplying by a conversion factor of 60 minutes per hour and dividing by 5,280 feet per mile.
- 56.651 rpm × (4π(7/3) feet/minute) × (60 minutes/hour) ÷ 5280 feet/mile = 38.157 mph (rounded to three decimal places).
Therefore, the vehicle is moving at a speed of approximately 38.2 miles per hour.