Two circles touch each other externally, the distance between their centres is 12 cm and the sum of their areas (in `cm^(2)`) is `74pi`. What is the radius of the smaller circle?

Options

A. 2.8
B. 4.5
C. 5
D. 3

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the radius of the smaller circle be \( r \) cm. Since the two circles touch each other externally, the radius of the larger circle will be \( 12 - r \) cm (because the distance between their centers is 12 cm).

### Step 2: Write the Area Equation
The area of the smaller circle is given by:
\[ \text{Area of smaller circle} = \pi r^2 \]
The area of the larger circle is given by:
\[ \text{Area of larger circle} = \pi (12 - r)^2 \]
According to the problem, the sum of their areas is \( 74\pi \):
\[ \pi r^2 + \pi (12 - r)^2 = 74\pi \]

### Step 3: Simplify the Equation
We can divide the entire equation by \( \pi \) (since \( \pi \) is a common factor):
\[ r^2 + (12 - r)^2 = 74 \]
Now, expand \( (12 - r)^2 \):
\[ r^2 + (144 - 24r + r^2) = 74 \]
Combine like terms:
\[ 2r^2 - 24r + 144 = 74 \]

### Step 4: Rearrange the Equation
Now, rearranging gives:
\[ 2r^2 - 24r + 144 - 74 = 0 \]
This simplifies to:
\[ 2r^2 - 24r + 70 = 0 \]

### Step 5: Divide by 2
To simplify, divide the entire equation by 2:
\[ r^2 - 12r + 35 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we will factor the quadratic:
\[ (r - 7)(r - 5) = 0 \]
This gives us two possible solutions for \( r \):
\[ r = 7 \quad \text{or} \quad r = 5 \]

### Step 7: Determine the Radius of the Smaller Circle
Since \( r \) represents the radius of the smaller circle, and we have defined \( r \) as the radius of the smaller circle, we take the smaller value:
\[ \text{Radius of the smaller circle} = 5 \text{ cm} \]

### Final Answer
Thus, the radius of the smaller circle is \( \boxed{5} \) cm.

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