The radius of a spherical diamond is measured as 6 cm with an error of 0.04 cm. Obtain the approximate error in calculating its volume. If the cost of `1cm^(3)` diamond is Rs 1600, what is the loss to the buyer of the diamond?

Answer

To solve the problem step by step, we will follow the process of calculating the approximate error in the volume of the spherical diamond and then determine the financial loss to the buyer.

### Step 1: Understand the formula for the volume of a sphere
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

### Step 2: Differentiate the volume with respect to the radius
To find the approximate error in volume due to an error in radius, we differentiate the volume with respect to \( r \):
\[ \frac{dV}{dr} = 4 \pi r^2 \]

### Step 3: Calculate the error in volume
Given:
- The radius \( r = 6 \) cm
- The error in radius \( dr = 0.04 \) cm

Now, substituting the values into the differentiated formula:
\[ dV = \frac{dV}{dr} \cdot dr = 4 \pi (6^2) \cdot 0.04 \]

Calculating \( 6^2 \):
\[ 6^2 = 36 \]

Now substituting this back into the equation:
\[ dV = 4 \pi (36) \cdot 0.04 \]

Calculating \( 4 \cdot 36 \):
\[ 4 \cdot 36 = 144 \]

Now substituting this value:
\[ dV = 144 \pi \cdot 0.04 \]

Calculating \( 144 \cdot 0.04 \):
\[ 144 \cdot 0.04 = 5.76 \]

Thus, we have:
\[ dV = 5.76 \pi \]

Using \( \pi \approx 3.14 \):
\[ dV \approx 5.76 \cdot 3.14 \approx 18.08 \, \text{cm}^3 \]

### Step 4: Calculate the financial loss
The cost of 1 cm³ of diamond is Rs 1600. Therefore, the loss to the buyer can be calculated as:
\[ \text{Loss} = \text{Volume Error} \times \text{Cost per cm}^3 \]
\[ \text{Loss} = 18.08 \times 1600 \]

Calculating the loss:
\[ \text{Loss} = 28928 \, \text{Rs} \]

### Final Answer
The approximate error in calculating the volume of the diamond is \( 18.08 \, \text{cm}^3 \) and the loss to the buyer is \( 28928 \, \text{Rs} \).

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