In the following find the approximate values, using differentials :
`sqrt(16.3)`

Answer

To find the approximate value of \(\sqrt{16.3}\) using differentials, we can follow these steps:

### Step 1: Define the function
Let \(y = \sqrt{x}\).

### Step 2: Differentiate the function
We need to find the derivative of \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \]

### Step 3: Choose a value for \(x\)
We will choose a value of \(x\) that is close to 16.3 and for which we know the square root. Here, we choose \(x = 16\) because \(\sqrt{16} = 4\).

### Step 4: Calculate \(dx\)
We need to find \(dx\) to represent the small change in \(x\):
\[ dx = 16.3 - 16 = 0.3 \]

### Step 5: Calculate \(dy\)
Now, we can calculate \(dy\) using the derivative:
\[ dy = \frac{dy}{dx} \cdot dx = \frac{1}{2\sqrt{16}} \cdot 0.3 \]
Substituting \(\sqrt{16} = 4\):
\[ dy = \frac{1}{2 \cdot 4} \cdot 0.3 = \frac{1}{8} \cdot 0.3 = \frac{0.3}{8} = \frac{3}{80} \]

### Step 6: Approximate the value of \(\sqrt{16.3}\)
Now, we can approximate \(\sqrt{16.3}\):
\[ \sqrt{16.3} \approx \sqrt{16} + dy = 4 + \frac{3}{80} \]
Calculating \(\frac{3}{80}\):
\[ \frac{3}{80} = 0.0375 \]
Thus,
\[ \sqrt{16.3} \approx 4 + 0.0375 = 4.0375 \]

### Final Result
Therefore, the approximate value of \(\sqrt{16.3}\) is:
\[ \sqrt{16.3} \approx 4.0375 \]