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Area of a square plot is 2304 ${{m}^{2}}$. Find the side of the square plot.
A. 48 m
B. 33.9 m
C. 39.9 m
D. 42 m

Last updated date: 13th Jun 2024
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Hint: To solve this question, we should know the relation between the length of the side of the square and the area of the square. The area of the square of side ‘s’ meter is given by the formula ${{s}^{2}}\text{ }{{m}^{2}}$. Using this relation, we get the side of the square as $s=\sqrt{2304}\text{ }m$ If $d={{a}^{x}}{{b}^{y}}{{c}^{z}}$ then square root of d is given by $\sqrt{d}={{a}^{\dfrac{x}{2}}}{{b}^{\dfrac{y}{2}}}{{c}^{\dfrac{z}{2}}}$. To get the value of the square root of 2304, we should do the prime factorization of 2304 and divide the powers of prime numbers by 2 to get the square root of the number.

Complete step-by-step solution:
In the question, we are given the area of the square plot as 2304 ${{m}^{2}}$ and asked to find the side of the square plot. We know that the relation between the area and side of the square is
Area $A={{s}^{2}}\text{ }{{m}^{2}}\to \left( 1 \right)$.
In the question A = 2304 ${{m}^{2}}$. Substituting the value of A in equation-1 we get
  & {{s}^{2}}=2304 \\
 & s=\sqrt{2304} \\
We should get the value of square root of 2304 to get the value of s. Using the factorisation method to get the square root of 2304, we get
  & 4\left| \!{\underline {\,
  2304 \,}} \right. \\
 & 4\left| \!{\underline {\,
  576 \,}} \right. \\
 & 4\left| \!{\underline {\,
  144 \,}} \right. \\
 & 4\left| \!{\underline {\,
  36 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
We can write that $2304={{4}^{4}}\times {{3}^{2}}$
If $d={{a}^{x}}{{b}^{y}}{{c}^{z}}$ then square root of d is given by $\sqrt{d}={{a}^{\dfrac{x}{2}}}{{b}^{\dfrac{y}{2}}}{{c}^{\dfrac{z}{2}}}$.
Using this property as d =2304, we get
$\sqrt{2304}={{4}^{\dfrac{4}{2}}}\times {{3}^{\dfrac{2}{1}}}={{4}^{2}}\times {{3}^{1}}=16\times 3=48$
So, the length of the side s = 48 m.
$\therefore $ The side of the square plot is 48 m and answer is option-A.

Note: We can do the problem in an alternative way by observing the options and given area. The number 2304 lies between 1600 and 2500 which are the squares of 40 and 50 respectively. So, the square root of 2304 also lies between 40 and 50. The last digit of 4 in 2304 is possible if the square root has the last digit as 2 or 8. So the two options for the answer are 42 and 48. By calculating the squares of 42 and 48, we get the answer. If 2304 is not a perfect square, the factorization procedure helps in getting the closest value.