Every time the pedals go through a `360^(@)` rotation on a certain bicycle, the tires rotate three times. If the tires are 24 inches in diameter, what is the minimum number of complete rotations of the pedals needed for the bicycle to travel at least 1 mile? (1mile=5,280feet)

Options

A. `24`
B. `281`
C. `561`
D. `5,280`

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the circumference of the tire.
The diameter of the tire is given as 24 inches. To find the circumference (C), we use the formula:
\[ C = \pi \times d \]
where \( d \) is the diameter.

First, convert the diameter from inches to feet:
\[ d = 24 \text{ inches} = \frac{24}{12} \text{ feet} = 2 \text{ feet} \]

Now, calculate the circumference:
\[ C = \pi \times 2 = 2\pi \text{ feet} \]

### Step 2: Determine the distance traveled in one complete rotation of the pedals.
We know that for every complete rotation of the pedals, the tires rotate three times. Therefore, the distance traveled for one complete rotation of the pedals is:
\[ \text{Distance} = 3 \times C = 3 \times 2\pi = 6\pi \text{ feet} \]

### Step 3: Convert the distance to be traveled (1 mile) into feet.
We know that:
\[ 1 \text{ mile} = 5280 \text{ feet} \]

### Step 4: Calculate the number of complete rotations of the pedals needed to travel 1 mile.
To find the number of rotations of the pedals required to travel 5280 feet, we divide the total distance by the distance covered in one rotation of the pedals:
\[ \text{Number of rotations} = \frac{5280 \text{ feet}}{6\pi \text{ feet}} \]

### Step 5: Perform the calculation.
Now we can calculate the number of rotations:
\[ \text{Number of rotations} = \frac{5280}{6\pi} \]

Using the approximate value of \( \pi \approx 3.14 \):
\[ \text{Number of rotations} \approx \frac{5280}{6 \times 3.14} \approx \frac{5280}{18.84} \approx 280.1 \]

### Step 6: Round to the nearest whole number.
Since we need the minimum number of complete rotations, we round up:
\[ \text{Minimum number of complete rotations} = 281 \]

Thus, the minimum number of complete rotations of the pedals needed for the bicycle to travel at least 1 mile is **281**.

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