Question

20 is divided into two parts so that the product of the cube of one quantity and the square of the other quantity is maximum. The parts are
A) $10,10$
B) $16,4$
C) $8,12$
D) $12,8$

Answer 1

Hint: In this problem, the given number twenty is divided into two parts. We have to find the two parts which divide the number twenty into three parts. Understanding the data and applying the differentiation we can solve the problem easily.
The formula used in the problem:
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

Complete step-by-step solution:
In this question, the given number twenty is divided into two parts.
We are going to find the two parts of the number twenty.
Let us consider that the two parts are $x,y$
That is the addition of two parts is twenty.
We can write this as, $x + y = 20$
and then, $y = 20 - x$
and then the given data is,
We can write this as,
 ${x^3} \times {y^2} = z$
Now substitute the value, we get
${x^3}{\left( {20 - x} \right)^2} = z$
Now apply the formula,
$z = {x^3}\left( {{{20}^2} - 2\left( {20} \right)x + {x^2}} \right)$
Now expand the term by using multiplication.
We get, $ = {x^3}\left( {400 - 40x + {x^2}} \right)$
$z = 400{x^3} - 40{x^4} + {x^5}$
Now differentiate the above term, we get
$\dfrac{{dz}}{{dx}} = 3 \times 400{x^2} - 4 \times 40{x^3} + 5{x^4}$
Now simplify the term, we get
$\dfrac{{dz}}{{dx}} = 1200{x^2} - 160{x^3} + 5{x^4}$
Now, let us take ${x^2}$as the common term. We get,
$ = {x^2}\left( {1200 - 160x + 5{x^2}} \right)$
Now rearrange the term, we get,
$\dfrac{{dz}}{{dx}} = {x^2}\left( {5{x^2} - 160x + 1200} \right)$
Now, $\dfrac{{dz}}{{dx}}$=0
Therefore, the factors of the quadratic equation are,
$x = 12,20$
Now do the second derivation, we get
$\dfrac{{{d^2}z}}{{d{x^2}}} = 2400x + 20{x^3} - 480{x^2}$
Substitute the value $x = 12$at the second derivation.
$\dfrac{{{d^2}z}}{{d{x^2}}} = 2400\left( {12} \right) + 20{\left( {12} \right)^3} - 480{\left( {12} \right)^2}$
Now simplify this we get,
$ = 28800 + 34560 - 69,120$
Now simplify this,
$\dfrac{{{d^2}z}}{{d{x^2}}} = - 5760$
After the simplification, we get a negative value and we know that when the second derivative is negative at any critical point then that point will give the maximum value of the function.
Hence the value of
$
  x = 12 \\
  y = 20 - 12 = 8 $
Therefore the answer is option D.
The explanation for option A:
From the above steps, we proved that option D is the answer.
Therefore option A is not an answer.
The explanation for option B:
From the complete solution, we proved that option D is the answer.
Therefore option B is not an answer.
The explanation for option C:
We proved that option A is the solution for the given question.
Therefore option C is not an answer.

Note: In mathematics, the derivative is the fundamental tool of calculus. There are many techniques of differentiation like product rule, quotient rule, chain rule, etc., in mathematics, the derivative shows an instantaneous rate of change.