Math Problem Statement
Solution
The problem states:
A rectangular pool measuring meters by meters is surrounded by a walkway of width meters. The question asks for the value of such that the area of the walkway is equal to the area of the pool.
Step-by-Step Solution:
-
Calculate the Area of the Pool:
- The pool is meters long and meters wide.
- Area of the pool square meters.
-
Calculate the Dimensions of the Pool + Walkway:
- The walkway surrounds the pool on all sides, adding meters to each dimension.
- So, the total length including the walkway is .
- The total width including the walkway is .
-
Calculate the Total Area of Pool + Walkway:
- The area of the pool and walkway combined is given by the area of a rectangle with dimensions .
- Expanding this:
-
Calculate the Area of the Walkway Alone:
- The area of the walkway is the total area of the pool and walkway minus the area of the pool.
- Thus, the area of the walkway is:
-
Set Up the Equation:
- We want the area of the walkway to equal the area of the pool:
-
Solve the Equation:
- Rearrange it into standard quadratic form:
- Divide by 4 to simplify:
- Factor the quadratic equation:
- So, or . Since a walkway width cannot be negative, meters.
Conclusion:
The value of that makes the area of the walkway equal to the area of the pool is 2 meters.
Would you like further details or explanations on any part of the solution?
Related Questions:
- How would the solution change if the pool were circular?
- What if the walkway's width varied along each side? How would we approach the problem then?
- How can you determine the perimeter of the walkway?
- How would the area change if the pool dimensions were doubled?
- What would the answer be if the walkway area were to be half the pool's area?
Tip:
When dealing with walkway or border problems, always consider the added dimensions carefully to set up equations correctly.
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Math Problem Analysis
Mathematical Concepts
Algebra
Geometry
Quadratic Equations
Formulas
Area of rectangle = length × width
Quadratic equation ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 8-10