The base and the altitude of triangular metal disc are 66 cm and 28 cm respectively. By drilling a circular hole through this metal disc, its area is reduced to one-third. Find the diameter of the hole. (Take only one side of the disc into consideration)

Answer

To solve the problem step by step, we will follow the outlined procedure to find the diameter of the circular hole drilled in the triangular metal disc.

### Step 1: Calculate the Area of the Triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Given:
- Base = 66 cm
- Height (altitude) = 28 cm

Substituting the values:
\[ A = \frac{1}{2} \times 66 \times 28 \]
Calculating this gives:
\[ A = \frac{1}{2} \times 1848 = 924 \text{ cm}^2 \]

### Step 2: Determine the Area Reduced by the Hole
The area of the triangle is reduced to one-third after drilling the hole. Therefore, the remaining area is:
\[ \text{Remaining Area} = \frac{1}{3} \times 924 = 308 \text{ cm}^2 \]
The area reduced by the hole is:
\[ \text{Area Reduced} = \text{Total Area} - \text{Remaining Area} = 924 - 308 = 616 \text{ cm}^2 \]

### Step 3: Set Up the Equation for the Area of the Circular Hole
The area of the circular hole can be expressed as:
\[ \text{Area of the hole} = \pi r^2 \]
We know from the previous step that this area is 616 cm², so:
\[ \pi r^2 = 616 \]

### Step 4: Solve for the Radius \( r \)
Using the value of \( \pi \) as \( \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 616 \]
To isolate \( r^2 \), multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 616 \times \frac{7}{22} \]
Calculating the right-hand side:
\[ r^2 = 28 \times 7 = 196 \]
Now, take the square root to find \( r \):
\[ r = \sqrt{196} = 14 \text{ cm} \]

### Step 5: Calculate the Diameter of the Hole
The diameter \( d \) of the hole is twice the radius:
\[ d = 2r = 2 \times 14 = 28 \text{ cm} \]

### Final Answer
The diameter of the hole is \( \boxed{28 \text{ cm}} \).

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