GED Mathematical Reasoning TestAlgebraic Problem Solving with Graphs and FunctionsMedium
A gardener is designing a rectangular flower bed. The length of the flower bed is 5 feet more than its width. If the area of the flower bed is 84 square feet, what is the width of the flower bed?
- A14 feet
- B17 feet
- C7 feet
- D12 feet
Show answer & explanationAnswer & explanation
Correct answer: C. 7 feet
Let 'w' be the width. Then the length 'l' is w + 5. The area is l * w, so (w + 5) * w = 84. This simplifies to w^2 + 5w - 84 = 0. Factoring the quadratic equation, we look for two numbers that multiply to -84 and add to 5. These are 12 and -7. So, (w + 12)(w - 7) = 0. The possible values for w are -12 and 7. Since width cannot be negative, w = 7 feet.
Why the other options are wrong
- A. If the width was 14, the length would be 19 and the area 266, which is too large.
- B. This would be the length if the width was 12. If width was 17, length would be 22 and area 374.
- D. This would be the length if the width was 7. If width was 12, length would be 17 and area 204.
Modeling Area with Quadratic Equations
This involves setting up a quadratic equation to represent the area of a geometric shape, typically a rectangle, when one dimension is expressed in terms of another, and the total area is known.
- Area of a rectangle = length × width.
- Express one dimension in terms of the other (e.g., L = W + x).
- Substitute into the area formula, resulting in a quadratic equation.
- Solve the quadratic equation and discard non-physical solutions (e.g., negative length/width).
Memory trick: Area of a 'Rect'angle is 'L' times 'W', often leading to 'Quad'ratics!