Garden Plot Quadratic Quote
A school improvement committee plans a rectangular herb garden beside the science wing. The plot’s length is 3 m more than twice its width, and the usable area is 65 m². A contractor says a 1 m wide paved border will run along all four sides outside the plot; the border’s outer edge must stay at least 1 m away from a drainage line that runs parallel to the longer side and is 9 m from the nearer longer side. The committee has ₹18,000 for paving at ₹450 per m² and wants to know whether the plan fits. A student sets the width as x metres and writes an equation from the area. Another member worries that one algebraic solution may be negative or may violate the drainage constraint. The committee also needs the outer dimensions of the paved border, not just the garden dimensions, before approving the work.
1. Form and solve the quadratic equation for the garden width. State the garden dimensions and explain why the other root is rejected.
Read the worked answer
Let the width be x m. Then the length is 2x+3 m.
Area equation: x(2x+3)=65 2x²+3x-65=0
Using the quadratic formula: D=3²-4(2)(-65)=9+520=529=23² x=(-3±23)/(2×2) x=(-3+23)/4=5 or x=(-3-23)/4=-6.5
The width must be positive, so x=5 m. The root -6.5 m is rejected because a length cannot be negative. Length=2(5)+3=13 m. Check: 5×13=65 m², so the dimensions are 5 m by 13 m.
2. Find the outer dimensions and paved area of the 1 m border, then decide whether ₹18,000 is enough at ₹450/m². Mention the drainage constraint.
Read the worked answer
The border adds 1 m on every side. Outer width=5+1+1=7 m Outer length=13+1+1=15 m Outer area=7×15=105 m² Garden area=65 m² Paved border area=105-65=40 m²
Cost=40×450=₹18,000. So the available money is exactly enough, with no contingency.
Drainage check: the border extends 1 m beyond each longer side. Its outer edge is 5+1=6 m from the nearer longer side. The drainage line is 9 m from that side, leaving 3 m clearance, so the required 1 m clearance is satisfied.