Question

Estimate the cube root of the number 40.
(a). 3.1
(b). 3.2
(c). 3.4
(d). 3.9

Answer 1

Hint: Write the number 40 as the sum of the nearest cube and an error term. Then use the method of approximation by differentials to find the answer.

Complete step by step answer:
A method for approximating the value of a function near a known value is called approximation by differentials. In this method, we first describe our function as cube root.

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

Then, we write the number 40 as a sum of the nearest cube and an error term. 40 can be written as 27 + 13. Then, we have:

\[x = 27............(2)\]

\[\Delta x = 13..............(3)\]

Using the method of approximation by differentials, we have:

\[f(x + \Delta x) = y + \Delta y...........(4)\]

We first calculate the value of y as follows:

\[y = \sqrt[3]{x}\]

\[y = \sqrt[3]{{27}}\]

The cube root of 27 is 3, hence, we have:

\[y = 3.............(5)\]

As a next step, we calculate the value of \[\Delta y\].

\[\Delta y = \dfrac{{dy}}{{dx}}\Delta x.............(6)\]

Now, we calculate the value of \[\dfrac{{dy}}{{dx}}\] as follows:

\[\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}(\sqrt[3]{x})\]

Differentiating \[\sqrt[3]{x}\], we have as follows:

\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{3{x^{\dfrac{2}{3}}}}}\]

Substituting the value of x as 27, we get:

\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{3{{(27)}^{\dfrac{2}{3}}}}}\]

Simplifying, we have:

\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{3(9)}}\]

\[\dfrac{{dy}}{{dx}} = \dfrac{1}{{27}}.............(7)\]

We now substitute the equation (7) in equation (6), we have:

\[\Delta y = \dfrac{1}{{27}}(13)\]

Simplifying the above equation, we have:

\[\Delta y = 0.48..............(8)\]

Substituting equation (8) and equation (5) in equation (4), we have:

\[f(27 + 13) = 3 + 0.48\]

Simplifying, we have:

\[f(40) = 3.48\]

The option that is closer to the obtained value is 3.4.

Hence, the correct answer is option (c).


Note: You can also simplify the cube root of 40 as \[2\sqrt[3]{5}\] and if you know the value of \[\sqrt[3]{5}\] which is 1.7, you can multiply with 2 to find the final answer.