Question

The approximate value of \(\sqrt[3]{64.04}\) is

• A. \(4.00043\)
• B. \(4.00076\)
• C. \(4.00083\)
• D. \(4.00064\)

Hint:

For approximation near \(a^3\), use: \[ \sqrt[3]{a^3+h}\approx a+\frac{h}{3a^2} \] This is much faster than direct calculation.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
We use differentials to find an approximate value. The formula is \(f(a+h) \approx f(a) + f'(a) \cdot h\).
Step 2: Key Formula or Approach:
Let \(f(x) = x^{1/3}\). Choose \(a = 64\) (perfect cube) and \(h = 0.04\).
Step 3: Detailed Explanation:
1. \(f(a) = f(64) = 64^{1/3} = 4\).
2. \(f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \cdot (x^{1/3})^2}\).
3. Evaluate \(f'(a)\) at \(a=64\):
\[ f'(64) = \frac{1}{3 \cdot 4^2} = \frac{1}{48} \] 4. Substitute into the approximation formula:
\[ \sqrt[3]{64.04} \approx 4 + \frac{1}{48} \cdot (0.04) \] \[ = 4 + \frac{4}{4800} = 4 + \frac{1}{1200} \] Calculating the decimal: \(\frac{1}{1200} = 0.000833\dots\)
So, \(\sqrt[3]{64.04} \approx 4.00083\).
Step 4: Final Answer:
The approximate value is \(4.00083\).