Question

A farmer is given a piece of land. He is given a rope of length \[400\] meters to declare the boundary of a rectangular field. What is the maximum area of land that he can get?
A. \[6400\] square meters
B. \[100\] square meters
C. \[64\] square meters
D. \[10000\] square meters

Answer 1

Hint: In the given question, we have been given the length of the rope which is used to mark the boundary of the rectangular field. This means that we have been given the perimeter of the rectangular field, which is \[400\] meters. Then, we have to find the maximum area of land which is possible with this given configuration (the sum of the sides of the rectangular field is \[400\]). We are going to solve this question by applying the formula of first order derivative \[\left( {f'x} \right)\] on the equation that is going to be formed. Then, we simplify the expression and put it equal to zero. Then we solve for the values of \[x\] and we are going to have our critical points for the expression. Then we just simply put in the critical values and the one giving off the maximum value is going to be our answer.

Formula Used:
We are going to use the formula of the first order derivative on the given expression and it is:
\[f'\left( {{x^n}} \right) = n{x^{n - 1}}\]

Complete answer:
In the given question, we have been given the perimeter of the rectangle.
Let the length and the breadth of the rectangle be represented by \[l\] and \[b\] respectively.
Then, \[2\left( {l + b} \right) = 400\]
or, \[l + b = 200\]
Now, \[b = 200 - l\] …(i)
Now, area of the rectangle is:
\[A = lb = \left( {200 - b} \right)b = 200b - {b^2}\]
Now, the first order derivative of \[A\] w.r.t. \[b\] is given by:
\[\dfrac{{dA}}{{db}} = \dfrac{{d\left( {200b - {b^2}} \right)}}{{dx}} = 200 - 2b\]
Now, we find the critical points of the expression by putting \[\dfrac{{dA}}{{db}} = 0\]
\[200 - 2b = 0\]
or \[b = 100\]
Now, putting \[b = 100\] in (i), we get:
\[l = 100\]
Hence, the area of the rectangular field with length and breadth each equal to \[100m\] is
\[A = 100 \times 100 = 10000\] square meters

Hence, the correct option is D).

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contains the known and the unknown and pick the one which is the most suitable for the answer. Then we put in the knowns into the formula, evaluate the answer and find the unknown. A thing to note and always remember is that it is important to see when to find the maximum of the critical points and when to find the minimum of them, which depends on what has been given in the given question.