Question

A rectangular farm has to be fenced on one long side, one short side and the diagonal. If the cost of fencing is Rs. 100 per meter, the area of the farm is $1200{{m}^{2}}$ and the short side is 30m long, how much would the job cost?
(a) Rs. 14000
(b) Rs. 12000
 (c) Rs. 7000
(d) Rs. 15000

Answer 1

Hint: First, we will assume the length of the rectangular farm be ‘l’ and its diagonal be d. Then, we use the formula of the area of the rectangle which is $A=l\times b$. Then, by substituting the value of breadth as 30 and area as 1200, we get the value of length. Then, by using the Pythagoras theorem which states that in any right angled triangle the square of the hypotenuse is equal to the sum of the square of the perpendicular and square of the base, we get the value of the diagonal length. Then, we will calculate the cost of fencing the one long side, one short side and the diagonal by multiplying it with 100 as given in the question.

Complete step by step answer:
In this question, we are supposed to find the cost of fencing on one long side, one short side and the diagonal.
So, let us assume the length of the rectangular farm be l ad its diagonal be d. Also, the breadth of the rectangular farm is given as 30m.

Now, by substituting the value of l=40m and b=30m, we get the value of diagonal as:
$\begin{align}
  & {{d}^{2}}={{40}^{2}}+{{30}^{2}} \\
 & \Rightarrow {{d}^{2}}=1600+900 \\
 & \Rightarrow {{d}^{2}}=2500 \\
 & \Rightarrow d=\sqrt{2500} \\
 & \Rightarrow d=50 \\
\end{align}$
Now, we want to fence the one long side, one short side and the diagonal. So, add all these sides as:
30+40+50=120mNow, by using the formula of the area(A) of the rectangle as:
$A=l\times 30$
So, the area A of the rectangle is given in the question as $1200{{m}^{2}}$ and by using it calculate the value of length l as:
$\begin{align}
  & 1200=l\times 30 \\
 & \Rightarrow l=\dfrac{1200}{30} \\
 & \Rightarrow l=40 \\
\end{align}$
So, we get the length of the rectangle as 40m.
Now, by using the Pythagoras theorem which states that in any right angled triangle the square of the hypotenuse is equal to the sum of the square of the perpendicular and square of the base.
So, we know that the rectangular farm has triangle which is which is at right angle and by using it we can calculate the value of the diagonal which is acting as hypotenuse as:
${{d}^{2}}={{l}^{2}}+{{b}^{2}}$
Then, we know from the question that the cost of fencing the 1m field is Rs. 100.
Then to find the cost of fencing the 120m, multiply it with 100 and get the desired amount as:
$100\times 120=12000$
So, the cost of fencing the one long side, one short side and the diagonal is Rs. 12000.
Hence, option (b) is correct.

Note: For this type of the question we must know some basic formulas and theorems so that we can easily proceed in them and calculate the desired results. In this question we require the formula of the area of the rectangle which is $A=l\times b$. Also , we should use some basic theorem as Pythagoras theorem which states that in any right angled triangle the square of the hypotenuse is equal to the sum of the square of the perpendicular and square of the base.