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  • Ishan wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same, but if the

Ishan wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same, but if the length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by 5300 m2. Using matrices, find the dimensions of the plot. Also, give the reason why he wants to donate the plot for a school.

 

 

 

 
 
 
 
 

Answers (1)

Let the and breadth of the land be x and y (in metres) respectively.

So,    (x-50)(y+50) = xy \Rightarrow 50x - 50y = 2500

i.e    x - y = 50\quad - (i)

And (x -10)(y-20) = xy - 5300 \Rightarrow -20x -10y = -5500

i.e    2x +y = 550\quad -(ii)

For solving (i) and (ii), let A = \begin{bmatrix} 1 & -1 \\ 2 & 1\end{bmatrix}, X = \begin{bmatrix}x \\y \end{bmatrix} & B = \begin{bmatrix} 50 \\550 \end{bmatrix}

\because AX = B \Rightarrow X = A^{-1}B \Rightarrow X = \frac{1}{1+2} \begin{bmatrix} 1 & 1 \\ -2 & 1\end{bmatrix}\begin{bmatrix} 50 \\550 \end{bmatrix} \quad \left[\because A^{-1} = \frac{1}{|A|}adjA \right ]

\Rightarrow X = \frac{1}{3} \begin{bmatrix} 50 + 550 \\ -100 + 550\end{bmatrix}= \begin{bmatrix} 200 \\150 \end{bmatrix}

\therefore x = 200 and y = 150

Hence dimensions of the land are: length = 200m & breath = 150 m

Reason:    Ishan knows the value of education, which may be the reason why he wishes to donate the plot for a school.

Posted by

Ravindra Pindel

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