Question

The dimension of a rectangular field are 50m by 40m. A flower bed is prepared inside this field leaving a gravel path of uniform width all around the flower bed. The total cost of laying flower bed and gravelling the path at Rs 30 and Rs 20 per square metres is Rs 52000. Find the width of the gravel path.

Answer 1

Hint: Now we have that Length and breadth of the rectangular field. Area of rectangle is given by Length × breadth. Now let us consider the width of gravel is x. Then we can say that length of the flower bed is 50 – 2x and breadth of the flower bed is 40 – 2x. hence we can find Area of the flower bed. Now Area of gravel path + Area of flower bed = Area of the total field. Hence we will have Area of flower bed as well as the area of gravel path. Now know the total cost of laying flower bed and gravelling the path at Rs 30 and Rs 20 per square metres is Rs 52000. Hence using this condition we will get an equation in x. Hence we will solve it to find the width of the gravel path.

Complete step by step answer:
Now we know that the dimension of the rectangular field is 50m by 40m

This means the length of a rectangular field is 50m.
And the breadth or rectangular field is 40m.
Now we know that area of the rectangle is given by length × breadth. Hence we can say
Area of a rectangular field is 50 × 40 = 200sq m ………………… (1)
Now Let us say that the width of the gravel is ‘x’ m
Hence we get the length of the flower bed is 50 – x – x = 50 – 2x.
Similarly, we have the breadth of flower bed is 40 – x – x = 40 – 2x.
Now Area of the flower bed is (50 – 2x) × (40 – 2x).
Area of flower bed is 2000 – 100x – 80x + $4{{x}^{2}}$
Hence area of the flower bed is $2000-180x+4{{x}^{2}}..............................(2)$
Also with the help of figure, we can say that.
Area of rectangle = Area of flower bed + Area of gravel
2000 = (50 – 2x) × (40 – 2x) + Area of gravel.
Area of gravel = 2000 – (50 – 2x) × (40 – 2x)
Area of gravel = 2000 – (2000 – 100x – 80x + $4{{x}^{2}}$)
Area of gravel = $2000-2000+100x+80x-4{{x}^{2}}$
Hence Area of gravel = $180x-4{{x}^{2}}..............................(3)$
Now we are given that cost of laying flower bed and gravelling the path is Rs 30 and Rs 20 per square metres.
Hence the cost of flower bed will be Area of flower bed × 30.
Cost of flower bed = $\left( 2000-180x+4{{x}^{2}} \right)\times 30...........(4)$
Also we have cost of gravel = $\left( 180x-4{{x}^{2}} \right)\times 20...........(5)$
Now the total cost is given as 52000, hence from equation (4) and equation (5) we have.
$52000=\left( 180x-4{{x}^{2}} \right)\times 20+\left( 2000-180x+4{{x}^{2}} \right)\times 30$
Now dividing the whole equation by 10 we get
$5200=\left( 180x-4{{x}^{2}} \right)\times 2+\left( 4{{x}^{2}}-180x+2000 \right)\times 3$
$\begin{align}
  & \Rightarrow 5200=360x-8{{x}^{2}}+12{{x}^{2}}-540x+6000 \\
 & \Rightarrow 5200=4{{x}^{2}}-180x+6000 \\
 & \Rightarrow 4{{x}^{2}}-180x+6000-5200=0 \\
 & \Rightarrow 4{{x}^{2}}-180x+800=0 \\
\end{align}$
Now dividing the whole equation by 4 we get
${{x}^{2}}-45x+200=0$
$\begin{align}
  & \Rightarrow {{x}^{2}}-5x-40x+200=0 \\
 & \Rightarrow x\left( x-5 \right)-40\left( x-5 \right)=0 \\
 & \Rightarrow \left( x-40 \right)\left( x-5 \right)=0 \\
\end{align}$
Hence x = 40 or x = 50.
Now we have the width of the rectangular field as 40 hence x cannot be 40.
Hence x = 5.
Hence the width of gravel is 5m.

Note:
Now note that if we consider the width of gravel to be x meters we have the length of the flower bed equal to the original length – 2x and not as original length – x, this is because x will be subtracted from the top as well as below. To understand this better follow the diagram.