Approximating the Cube Root of 0.99

The question asks us to find the value closest to the cube root of 0.99, denoted as $ \sqrt[3]{0.99} $. We need to compare this value to the given options: 0.33, 0.14, 0.99, and 0.45.

Intuitive Estimation

We know that the cube root of 1 is 1 ($ \sqrt[3]{1} = 1 $). Since 0.99 is very close to 1, its cube root should also be very close to 1. Let's examine the options:

• 0.33 is much smaller than 1.
• 0.14 is much smaller than 1.
• 0.99 is very close to 1.
• 0.45 is much smaller than 1.

Based on this simple estimation, 0.99 appears to be the most likely answer.

Using Calculus for Approximation

We can use linear approximation to get a more precise estimate. Let the function be $ f(x) = \sqrt[3]{x} = x^{1/3} $. We want to approximate $ f(0.99) $. We can use the point $ a=1 $ where the function value and its derivative are easy to calculate.

The linear approximation formula is:

$ f(x) \approx f(a) + f'(a)(x-a) $

1. Calculate the function value at a:

   $ f(1) = \sqrt[3]{1} = 1 $

2. Find the derivative of the function:

   $ f'(x) = \frac{d}{dx}(x^{1/3}) = \frac{1}{3}x^{(1/3 - 1)} = \frac{1}{3}x^{-2/3} $

3. Calculate the derivative value at a:

   $ f'(1) = \frac{1}{3}(1)^{-2/3} = \frac{1}{3}(1) = \frac{1}{3} $

4. Apply the linear approximation formula with $ x = 0.99 $ and $ a = 1 $:

   $ f(0.99) \approx f(1) + f'(1)(0.99 - 1) $

   $ f(0.99) \approx 1 + \frac{1}{3}(0.99 - 1) $

   $ f(0.99) \approx 1 + \frac{1}{3}(-0.01) $

   $ f(0.99) \approx 1 - \frac{0.01}{3} $

   $ f(0.99) \approx 1 - 0.00333... $

   $ f(0.99) \approx 0.99667 $

Comparing the Approximation with Options

Our calculated approximation is $ \approx 0.99667 $. Let's see which option is closest to this value:

The smallest difference is 0.00667, which corresponds to option 3 (0.99).

Conclusion

Both the intuitive approach and the linear approximation method show that 0.99 is the value closest to $ \sqrt[3]{0.99} $ among the given choices.