Problem 116 - Algebra and Trigonometry Real Mathematics, Real People

Question

The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4 inches, 2 inches, and 14 inches, respectively. A cyclist is pedaling at a rate of 1 revolution per second. (a) Find the speed of the bicycle in feet per second and miles per hour. (b) Use your result from part (a) to write a function for the distance \(d\) (in miles) a cyclist travels in terms of the number \(n\) of revolutions of the pedal sprocket. (c) Write a function for the distance \(d\) (in miles) a cyclist travels in terms of time \(t\) (in seconds). Compare this function with the function from part (b).

Step-by-Step Solution

Answer

Speed of the bicycle is approximately 5.86 feet per second or approximately 4 miles per hour. Function for distance based on the number of revolutions is \(d = n* 0.000757106\) miles, and function for distance based on time is \(d = t* 0.001136364\) miles.

Step 1: Calculate the Speed

First, determine the speed of the bicycle. This requires calculating the circumference of the wheel, which is given by the formula \(C = 2\pi r\), where \(r\) is the radius of the wheel. Since the radius of the wheel is 14 inches, the circumference will be \(2\pi *14 \) inches. Since we are asked to find the speed in feet, we must convert inches to feet, knowing that 1 foot equals 12 inches. The number of wheel revolutions per pedal revolution is the ratio of the pedal sprocket radius to the wheel sprocket radius. Multiply the wheel circumference by the number of pedal revolutions per second and by the number of wheel revolutions per pedal revolution to obtain the speed in feet/second.

Step 2: Develop Function for Distance Based on Number of Revolutions

To find a function for the distance travelled in terms of the number of revolutions, multiply the distance travelled per pedal revolution by the number of pedal revolutions. Since the distance travelled per pedal revolution is the circumference of the wheel times the number of wheel revolutions per pedal revolution, the function becomes \(d = n* C * (\text{{pedal sprocket radius}}/ \text{{wheel sprocket radius}})\). Convert the distance d into miles by knowing that 1 mile is 5280 feet.

Step 3: Develop Function for Distance Based on Time

To find a function for the distance travelled in terms of time, multiply the speed of the bicycle (distance travelled per second) by time. Convert distance into miles by knowing that 1 mile equals 5280 feet. This function should be compared with the function from Step 2. In both functions, the distance travelled is directly proportional to either the number of pedal revolutions or the time duration, and the constants in both functions are the same, indicating that these functions represent the same relationship.

Key Concepts

Circumference Calculation

To calculate the circumference of a circle, we use the formula: the circumference \(C = 2\pi r\), where \(r\) is the radius of the circle. For a bicycle wheel with a radius of 14 inches, the circumference becomes \(2\pi \times 14\) inches. Understanding this formula helps in determining how far a wheel travels with each full revolution. The circumference represents the complete path the wheel covers in one spin.

• The unit for circumference will initially be in inches due to the given radius in inches.
• To easily work with other measurements like feet, it’s essential to convert these units later on.

Knowing the circumference is crucial because it links the rotational movement of the wheel to linear distance traveled, helping us calculate speed effectively.

Revolutions Per Second

Revolutions per second (RPS) is a measure of how many full turns or spins an object makes in one second. In the context of biking, it describes how often the pedals or wheels make a full circle in a second. For example, if the cyclist pedals at a rate of 1 revolution per second, this means the pedals complete a full circle every second.

• This rate is crucial for determining the distance traveled since it shows how active the bike’s systems are.
• It allows us to map out how many full wheel rotations occur in tandem with each pedal rotation based on the gear ratios.

By calculating the revolutions per second, we can estimate how the mechanical systems work together to propel the bike over various distances.

Distance Function

The distance function describes how far a bicycle travels and it can be calculated using different variables, such as the number of revolutions or the time spent riding. If we take the number of pedal revolutions \(n\) and the circumference of the wheel \(C\), the distance function can be expressed as:\[d = n \times C \times \left(\frac{\text{pedal sprocket radius}}{\text{wheel sprocket radius}}\right)\]This formula shows that the distance \(d\) covered depends on the number of pedal revolutions and the effective circumference that results from gearing.

• It accounts for gear ratios by including the ratio of the pedal sprocket radius to the wheel sprocket radius.
• Understanding this concept is key to designing efficient gear systems for bicycles.

Unit Conversion

Unit conversion is a pivotal skill in solving problems involving measurements. Bicycle speed is typically sought in feet per second or miles per hour. Hence, converting units properly ensures accurate and meaningful results. To convert from inches to feet, recall that 12 inches equals 1 foot. Similarly, to convert from feet to miles, use the conversion: 5280 feet equals 1 mile.

• Starting with the wheel’s circumference in inches, divide by 12 to switch to feet.
• To find the speed in miles per hour, calculate the speed in feet per second, then convert to miles per hour accordingly.

Mastering these conversions lets you toggle between different systems of measurement, making your answers relevant and useful across various contexts. This is especially important in real-world applications where units vary.