The approximate value of `root(3)(28)` is

Options

A. `3.038`
B. `3.035`
C. `3.036`
D. `3.037`

Answer

To find the approximate value of \( \sqrt[3]{28} \), we can use the concept of derivatives. We will use the function \( f(x) = x^{1/3} \) and apply the linear approximation formula.

### Step 1: Define the function and its derivative
Let \( f(x) = x^{1/3} \). We need to find the derivative \( f'(x) \).

\[ f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}} \]

### Step 2: Choose a point close to 28
We know that \( 27 \) is a perfect cube, so we will take \( x = 27 \) (since \( \sqrt[3]{27} = 3 \)) and \( h = 1 \) (because \( 28 = 27 + 1 \)).

### Step 3: Use the linear approximation formula
The linear approximation formula is given by:

\[ f(x + h) \approx f(x) + f'(x) \cdot h \]

Substituting \( x = 27 \) and \( h = 1 \):

\[ f(28) \approx f(27) + f'(27) \cdot 1 \]

### Step 4: Calculate \( f(27) \)
We know:

\[ f(27) = 27^{1/3} = 3 \]

### Step 5: Calculate \( f'(27) \)
Now we need to find \( f'(27) \):

\[ f'(27) = \frac{1}{3 \sqrt[3]{27^2}} = \frac{1}{3 \cdot 9} = \frac{1}{27} \]

### Step 6: Substitute back into the approximation
Now we can substitute back into the linear approximation formula:

\[ f(28) \approx 3 + \frac{1}{27} \cdot 1 = 3 + \frac{1}{27} \]

### Step 7: Calculate \( \frac{1}{27} \)
Calculating \( \frac{1}{27} \):

\[ \frac{1}{27} \approx 0.037 \]

### Step 8: Final approximation
Thus, we have:

\[ f(28) \approx 3 + 0.037 = 3.037 \]

Therefore, the approximate value of \( \sqrt[3]{28} \) is \( \approx 3.037 \).