A rectangular plot has a length that is 6 meters more than its width. If the area of the plot is 216 square meters, what is the perimeter of the plot?
- A60 meters
- B54 meters
- C72 meters
- D66 meters
Show answer & explanationAnswer & explanation
Correct answer: A. 60 meters
Let the width of the rectangular plot be 'w' meters. Then the length 'l' is w + 6 meters. Area of a rectangle = length × width, so (w + 6) × w = 216. w^2 + 6w = 216. w^2 + 6w - 216 = 0. We need to solve this quadratic equation for 'w'. We look for two numbers that multiply to -216 and add to 6. These numbers are 18 and -12. So, (w + 18)(w - 12) = 0. Possible values for w are -18 or 12. Since width cannot be negative, w = 12 meters. Length l = w + 6 = 12 + 6 = 18 meters. Perimeter of a rectangle = 2 × (length + width) = 2 × (18 + 12) = 2 × 30 = 60 meters.
Why the other options are wrong
- B. Incorrect. This might arise from miscalculating the dimensions or the perimeter.
- C. Incorrect. This might arise from miscalculating the dimensions or the perimeter.
- D. Incorrect. This might arise from miscalculating the dimensions or the perimeter.
Quadratic Equations (Geometric Problems)
Geometric problems often lead to quadratic equations when relationships between dimensions (like length and width) are described, and the area or another quadratic property is given. Solving the quadratic equation yields the possible dimensions.
- Formulate an equation relating dimensions and area (or other property).
- This often results in a quadratic equation of the form ax² + bx + c = 0.
- Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.
- Discard negative or unrealistic solutions for physical dimensions.
Memory trick: Dimensions' dance, quadratic chance, to find the area's expanse!