The sum of the circumferences of two circles which touch each other externally is 176 cm. What is the ratio of the radius of the larger circle to that of the smaller circle, if the sum of squares of the radii of the circles is 400 cm ?

Options

A. `7 : 8`
B. `8 : 7`
C. `3 : 4`
D. `4 : 3`

Answer

To solve the problem step by step, we will follow the logic laid out in the video transcript.

### Step 1: Define the Variables
Let \( r_1 \) be the radius of the larger circle and \( r_2 \) be the radius of the smaller circle.

### Step 2: Set Up the First Equation
The sum of the circumferences of the two circles is given as 176 cm. The circumference of a circle is given by the formula \( 2\pi r \). Therefore, we can write:
\[ 2\pi r_1 + 2\pi r_2 = 176 \]
Factoring out \( 2\pi \):
\[ 2\pi (r_1 + r_2) = 176 \]
Dividing both sides by \( 2\pi \):
\[ r_1 + r_2 = \frac{176}{2\pi} = \frac{88}{\pi} \]

### Step 3: Set Up the Second Equation
We are also given that the sum of the squares of the radii is 400 cm:
\[ r_1^2 + r_2^2 = 400 \]

### Step 4: Square the First Equation
Now, we will square the first equation:
\[ (r_1 + r_2)^2 = \left(\frac{88}{\pi}\right)^2 \]
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ r_1^2 + r_2^2 + 2r_1r_2 = \left(\frac{88}{\pi}\right)^2 \]

### Step 5: Substitute the Known Value
We know from the second equation that \( r_1^2 + r_2^2 = 400 \). Substitute this into the equation:
\[ 400 + 2r_1r_2 = \left(\frac{88}{\pi}\right)^2 \]

### Step 6: Calculate \( \left(\frac{88}{\pi}\right)^2 \)
Calculating \( \left(\frac{88}{\pi}\right)^2 \):
\[ \left(\frac{88}{\pi}\right)^2 = \frac{7744}{\pi^2} \]
Now, we can rewrite the equation:
\[ 400 + 2r_1r_2 = \frac{7744}{\pi^2} \]

### Step 7: Solve for \( r_1r_2 \)
Rearranging gives:
\[ 2r_1r_2 = \frac{7744}{\pi^2} - 400 \]
\[ r_1r_2 = \frac{1}{2} \left(\frac{7744}{\pi^2} - 400\right) \]

### Step 8: Use the Difference of Squares
We can also use the identity \( r_1^2 - r_2^2 = (r_1 - r_2)(r_1 + r_2) \) to find \( r_1 - r_2 \):
\[ r_1 - r_2 = \sqrt{(r_1 + r_2)^2 - 4r_1r_2} \]
Substituting the known values:
\[ r_1 - r_2 = \sqrt{\left(\frac{88}{\pi}\right)^2 - 4 \cdot \frac{1}{2} \left(\frac{7744}{\pi^2} - 400\right)} \]

### Step 9: Solve for \( r_1 \) and \( r_2 \)
Now we have two equations:
1. \( r_1 + r_2 = \frac{88}{\pi} \)
2. \( r_1 - r_2 = \sqrt{16} = 4 \)

Adding these two equations:
\[ 2r_1 = \frac{88}{\pi} + 4 \]
Thus,
\[ r_1 = \frac{44}{\pi} + 2 \]

Subtracting the second from the first:
\[ 2r_2 = \frac{88}{\pi} - 4 \]
Thus,
\[ r_2 = \frac{44}{\pi} - 2 \]

### Step 10: Calculate the Ratio
Finally, the ratio of the radius of the larger circle to that of the smaller circle is:
\[ \frac{r_1}{r_2} = \frac{\frac{44}{\pi} + 2}{\frac{44}{\pi} - 2} \]

### Final Ratio
After simplifying, we find:
\[ \frac{r_1}{r_2} = \frac{16}{12} = \frac{4}{3} \]