Find the diameter of a wheel that makes 113 revolutions to go 2 km 26 decameters. (Take `pi = (22)/(7)`)

Options

A. `4(4)/(13)m`
B. `6(4)/(11)m`
C. `12(4)/(11)m`
D. `12(8)/(11)m`

Answer

To find the diameter of a wheel that makes 113 revolutions to travel a distance of 2 km 26 decameters, we will follow these steps:

### Step 1: Convert the total distance traveled into meters.
- We know that 1 kilometer = 1000 meters and 1 decameter = 10 meters.
- Therefore, 2 kilometers = 2 × 1000 = 2000 meters.
- And 26 decameters = 26 × 10 = 260 meters.
- Now, we add these two distances together:
\[ \text{Total distance} = 2000 \text{ meters} + 260 \text{ meters} = 2260 \text{ meters}. \]

### Step 2: Calculate the distance traveled in one revolution.
- The distance traveled in one revolution of the wheel is equal to the circumference of the wheel.
- The formula for the circumference (C) of a circle is given by:
\[ C = \pi \times D, \]
where \(D\) is the diameter of the wheel.

### Step 3: Set up the equation for the total distance traveled.
- Since the wheel makes 113 revolutions, the total distance traveled can also be expressed as:
\[ \text{Total distance} = \text{Number of revolutions} \times \text{Circumference}. \]
- Therefore, we have:
\[ 2260 = 113 \times \left(\frac{22}{7} \times D\right). \]

### Step 4: Solve for the diameter (D).
- Rearranging the equation gives:
\[ D = \frac{2260 \times 7}{113 \times 22}. \]
- Now, calculate the right-hand side:
\[ D = \frac{15820}{2486}. \]
- Simplifying this fraction:
\[ D = 6 + \frac{4}{11} \text{ meters}. \]

### Final Answer:
The diameter of the wheel is \(6 \frac{4}{11}\) meters.

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