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

Concept:
Use approximation by differentials. Let \[ y=x^{1/3} \] Then \[ dy=\frac{1}{3x^{2/3}}dx \] ip

Step 1: Choose the nearby exact cube.
Since \[ 64=4^3, \] we take \[ x=64,\qquad dx=64.04-64=0.04 \] ip

Step 2: Find \(dy\).
\[ dy=\frac{1}{3(64)^{2/3}}dx \] Now, \[ (64)^{1/3}=4 \quad \Rightarrow \quad (64)^{2/3}=16 \] So, \[ dy=\frac{1}{3\cdot 16}(0.04)=\frac{0.04}{48} \] \[ dy=0.000833\ldots \] ip

Step 3: Find the approximate cube root.
\[ \sqrt[3]{64.04}\approx 4+0.000833=4.00083 \] ip Hence, the correct answer is:
\[ \boxed{(C)\ 4.00083} \]