Mr. Chaudhary has a rectangular pond for fish farming. The pond covers 8000 sq. m. of his land and its perimeter is 360 m.
(i) Find the length and breadth of the pond.
(ii) By what percent should the shorter edge of the pond be increased to make it a square?
Answer (2)
Set up equations for the area and perimeter of the rectangle: l b = 8000 and 2 ( l + b ) = 360 .
Solve the system of equations to find the length and breadth: l = 100 m and b = 80 m.
Calculate the increase needed to make the shorter side equal to the longer side: 100 − 80 = 20 m.
Determine the percentage increase: 80 20 × 100 = 25% . The final answer is 25% .
Explanation
Problem Analysis Let's analyze the problem. We have a rectangular pond with a known area and perimeter. We need to find the length and breadth of the pond and then determine the percentage increase required for the shorter side to make it a square.
Setting up Equations Let l be the length and b be the breadth of the rectangular pond. We are given that the area A = l b = 8000 sq. m and the perimeter P = 2 ( l + b ) = 360 m. From the perimeter equation, we can simplify it to l + b = 180 .
Solving the System of Equations Now we have a system of two equations:
l × b = 8000
l + b = 180
We can solve for l in the second equation: l = 180 − b . Substituting this into the first equation, we get ( 180 − b ) b = 8000 .
Applying the Quadratic Formula Expanding and rearranging the equation, we get a quadratic equation: 180 b − b 2 = 8000 , which can be rewritten as b 2 − 180 b + 8000 = 0 . We can solve this quadratic equation for b using the quadratic formula: b = 2 ( 1 ) − ( − 180 ) ± ( − 180 ) 2 − 4 ( 1 ) ( 8000 )
Finding Possible Breadths Calculating the discriminant: ( − 180 ) 2 − 4 ( 1 ) ( 8000 ) = 32400 − 32000 = 400 So, b = 2 180 ± 400 = 2 180 ± 20 This gives us two possible values for b : b 1 = 2 180 + 20 = 2 200 = 100 b 2 = 2 180 − 20 = 2 160 = 80
Determining Length and Breadth If b = 100 , then l = 180 − 100 = 80 . If b = 80 , then l = 180 − 80 = 100 . So, the length and breadth of the pond are 100 m and 80 m, respectively.
Calculating the Increase The shorter edge of the pond is 80 m, and the longer edge is 100 m. To make the pond a square, the shorter edge must be increased to match the length of the longer edge. The increase required is 100 − 80 = 20 m.
Calculating Percentage Increase The percentage increase required is: s h or t er e d g e in cre a se × 100 = 80 20 × 100 = 25% Therefore, the shorter edge should be increased by 25% to make the pond a square.
Final Answer The length and breadth of the pond are 100 m and 80 m, respectively. The shorter edge should be increased by 25% to make the pond a square.
Examples
Imagine you are designing a garden and want to include a rectangular pond. You know the total area you want the pond to cover and the amount of fencing you have for the perimeter. This problem helps you determine the dimensions of the pond. Furthermore, if you decide you want a square pond instead, you can calculate how much you need to increase the shorter side. This type of problem is useful in landscape design, construction, and resource management.
The length and breadth of the pond are 100 m and 80 m, respectively. To transform the shorter edge into a square, it needs to be increased by 25%.
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