Difference between circumference of a circular vessel (candy maker) and its diameter is 60 cm. If 40 cm2 is waste material and 16 square boxes required to pack remaining quantity of candy, then find side of each square box?

Options

A. 6 cm
B. 8 cm
C. 10 cm
D. 5 cm

Answer

To solve the problem step by step, we will follow the given information and derive the necessary values.

### Step 1: Understand the relationship between circumference and diameter
The problem states that the difference between the circumference of a circular vessel and its diameter is 60 cm.

Let:
- \( C \) = Circumference of the circular vessel
- \( D \) = Diameter of the circular vessel

We know that:
- \( C = 2\pi r \) (where \( r \) is the radius)
- \( D = 2r \)

According to the problem:
\[ C - D = 60 \]

### Step 2: Substitute the formulas for circumference and diameter
Substituting the formulas into the equation gives:
\[ 2\pi r - 2r = 60 \]

### Step 3: Factor out the common term
Factoring out \( 2r \) from the left side:
\[ 2r(\pi - 1) = 60 \]

### Step 4: Solve for \( r \)
Dividing both sides by \( 2(\pi - 1) \):
\[ r = \frac{60}{2(\pi - 1)} = \frac{30}{\pi - 1} \]

### Step 5: Calculate the approximate value of \( r \)
Using \( \pi \approx 3.14 \):
\[ r \approx \frac{30}{3.14 - 1} = \frac{30}{2.14} \approx 14.02 \, \text{cm} \]

### Step 6: Calculate the area of the circular vessel
The area \( A \) of the circular vessel is given by:
\[ A = \pi r^2 \]

Substituting the value of \( r \):
\[ A \approx \pi (14.02)^2 \approx 3.14 \times 196.56 \approx 616 \, \text{cm}^2 \]

### Step 7: Calculate the useful material
The problem states that 40 cm² is waste material. Therefore, the useful material is:
\[ \text{Useful Material} = 616 - 40 = 576 \, \text{cm}^2 \]

### Step 8: Determine the volume of one square box
Since 16 square boxes are required to pack the remaining quantity of candy, the area of one square box is:
\[ \text{Area of one box} = \frac{576}{16} = 36 \, \text{cm}^2 \]

### Step 9: Find the side of each square box
Let the side of each square box be \( a \). Then:
\[ a^2 = 36 \]

Taking the square root of both sides:
\[ a = 6 \, \text{cm} \]

### Final Answer
The side of each square box is **6 cm**.

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