The value of a coin varies to the square of its radius, when its thickness is constant. The radius of a coin is 1.5 cm and its value is Rs. 2. What will be radius of a coin if its value is Rs 5?

Answer

To solve the problem, we need to use the relationship between the value of the coin and the square of its radius. Here’s a step-by-step solution:

### Step 1: Understand the relationship
The value \( V \) of the coin varies with the square of its radius \( r \). This can be expressed mathematically as:
\[ V \propto r^2 \]
This means that:
\[ V = k \cdot r^2 \]
where \( k \) is a constant of proportionality.

### Step 2: Find the constant \( k \)
We know that when the radius \( r = 1.5 \) cm, the value \( V = 2 \) Rs. We can use this information to find \( k \):
\[ 2 = k \cdot (1.5)^2 \]
Calculating \( (1.5)^2 \):
\[ (1.5)^2 = 2.25 \]
Now substituting back into the equation:
\[ 2 = k \cdot 2.25 \]
To find \( k \), divide both sides by \( 2.25 \):
\[ k = \frac{2}{2.25} = \frac{8}{9} \]

### Step 3: Set up the equation for the new value
Now we need to find the radius \( r \) when the value \( V = 5 \) Rs. Using the formula:
\[ V = k \cdot r^2 \]
Substituting the known values:
\[ 5 = \frac{8}{9} \cdot r^2 \]

### Step 4: Solve for \( r^2 \)
To isolate \( r^2 \), multiply both sides by \( \frac{9}{8} \):
\[ r^2 = 5 \cdot \frac{9}{8} \]
Calculating the right side:
\[ r^2 = \frac{45}{8} \]

### Step 5: Find \( r \)
Now, take the square root of both sides to find \( r \):
\[ r = \sqrt{\frac{45}{8}} \]
This can be simplified:
\[ r = \frac{\sqrt{45}}{\sqrt{8}} = \frac{3\sqrt{5}}{2\sqrt{2}} \]
Rationalizing the denominator:
\[ r = \frac{3\sqrt{10}}{4} \]

### Step 6: Approximate the value of \( r \)
To get a numerical approximation:
\[ \sqrt{10} \approx 3.162 \]
Thus:
\[ r \approx \frac{3 \cdot 3.162}{4} \approx \frac{9.486}{4} \approx 2.372 \text{ cm} \]

### Final Answer
The radius of the coin when its value is Rs. 5 is approximately **2.37 cm**.

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