Question

The area of a rectangular garden, ABCD, is 100 m\[^2\]. Inside the garden, there is a rectangular lawn, EFGH, whose sides are parallel to those of the garden. Find the area of the lawn, EFGH (in sq. meters).

A. \[110 - 5x - \dfrac{{200}}{x}\]
B. \[110 + 5x - \dfrac{{200}}{x}\]
C. \[110 + 5x + \dfrac{{200}}{x}\]
D. \[110 - 5x + \dfrac{{200}}{x}\]

Answer 1

Hint: First, we will find the dimension of the rectangle. Then use the formula of area of the rectangle is \[A = lb\], where \[l\] is the length and \[b\] is the breadth. Apply this formula of area of the rectangle, and then use the given conditions to find the required value.

Complete step by step Answer:

We are given that the area \[A\] of \[{\text{ABCD}}\] is 100 m\[^2\] and the length \[l\] of \[{\text{ABCD}}\] is \[x\] m in the given figure.

We know that the area of the rectangle is calculated as \[A = lb\], where \[l\] is the length of the rectangle and \[b\] is the breadth of the rectangle.

Substituting the value of the area \[A\] of the rectangle \[{\text{ABCD}}\] and \[l\] in the above formula of the area from the given figure, we get

\[100 = xb\]

Dividing the above equation by the length of the rectangle \[x\] on each of the sides, we get

\[
   \Rightarrow \dfrac{{100}}{x} = \dfrac{{xb}}{x} \\
   \Rightarrow b = \dfrac{{100}}{x} \\
 \]

We will now find the length for the rectangle \[{\text{EFGH}}\] from the given condition of the given figure of the lawn.

\[x - \left( {1 + 1} \right) = x - 2{\text{ m}}\]

Now we will now find the breadth of the rectangle \[{\text{EFGH}}\] from the given conditions of the given figure of the lawn.

\[\dfrac{{100}}{x} - \left( {4 + 1} \right) = \dfrac{{100}}{x} - 5{\text{ m}}\]

Using the formula of area of the rectangle \[A = lb\] for the rectangle \[{\text{EFGH}}\] in the given figure, we get

\[
   \Rightarrow \left( {x - 2} \right)\left( {\dfrac{{100}}{x} - 5} \right) \\
   \Rightarrow x\left( {\dfrac{{100}}{x}} \right) - 5x - 2\left( {\dfrac{{100}}{x}} \right) + 10 \\
   \Rightarrow 100 - 5x - \dfrac{{200}}{x} + 10 \\
   \Rightarrow 110 - 5x - \dfrac{{200}}{x} \\
 \]

Thus, the area of the rectangular lawn, EFGH is \[110 - 5x - \dfrac{{200}}{x}\].

Hence, option A will be correct.

Note: In solving these types of questions, you should be familiar with the formula of the area of the rectangle. One should remember to subtract the length of the rectangle from the area of the rectangle ABCD to find the required side or else the final answer will be wrong. Some students use the formula of the perimeter of the rectangle instead of the area of the rectangle, which is wrong.