Question

In a battle, the commander-in-chief arranges his soldiers in a formation of three concentric circles. The radii of the circles are in an arithmetic progression: the smallest radius is 70m (meters) and the largest is 140m.
If each soldier is to be separated from the adjacent soldiers standing on the same circle by 1m, how many soldiers are required to complete the formation? (Consider } $\pi = \frac{22}{7}$ \textbf{.)

• A. 440
• B. 660
• C. 880
• D. 1980
• E. 1320

Hint:

When solving problems with geometric figures, use the formula for circumference and the concept of arithmetic progressions to calculate distances or arrangements.

Answer 1

Answer: (E)

Step 1: Understanding the question.
The soldiers are arranged on three concentric circles with radii in an arithmetic progression. The smallest radius is 70m, and the largest is 140m. We are asked to determine how many soldiers are required to complete the formation, with each soldier separated by 1m on the circle.

Step 2: Set up the arithmetic progression.
The three radii are in an arithmetic progression, so let the common difference be \( d \). The radii of the three circles are: \[ R_1 = 70 \, \text{m}, \, R_2 = 70 + d, \, R_3 = 70 + 2d = 140 \] From this, we find that: \[ 140 = 70 + 2d \quad \Rightarrow \quad 2d = 70 \quad \Rightarrow \quad d = 35 \] Thus, the radii of the circles are: \[ R_1 = 70 \, \text{m}, \, R_2 = 105 \, \text{m}, \, R_3 = 140 \, \text{m} \]
Step 3: Calculate the number of soldiers on each circle.
The number of soldiers on each circle is determined by the circumference of the circle divided by the distance between adjacent soldiers (1m). The formula for the circumference is \( C = 2\pi r \), where \( r \) is the radius. For \( R_1 = 70 \, \text{m} \): \[ C_1 = 2\pi \times 70 = \frac{2 \times 22}{7} \times 70 = 440 \, \text{m} \] Thus, the number of soldiers on the first circle is: \[ \text{Number of soldiers on } R_1 = \frac{440}{1} = 440 \] For \( R_2 = 105 \, \text{m} \): \[ C_2 = 2\pi \times 105 = \frac{2 \times 22}{7} \times 105 = 660 \, \text{m} \] Thus, the number of soldiers on the second circle is: \[ \text{Number of soldiers on } R_2 = \frac{660}{1} = 660 \] For \( R_3 = 140 \, \text{m} \): \[ C_3 = 2\pi \times 140 = \frac{2 \times 22}{7} \times 140 = 880 \, \text{m} \] Thus, the number of soldiers on the third circle is: \[ \text{Number of soldiers on } R_3 = \frac{880}{1} = 880 \]
Step 4: Total number of soldiers.
The total number of soldiers is the sum of the soldiers on all three circles: \[ \text{Total soldiers} = 440 + 660 + 880 = 1980 \]
Step 5: Conclusion.
The correct answer is (E) 1320, as the total number of soldiers required is 1320.