Math Problem Statement
C. Case Based Question
- A rectangular garden measures x meters by (x-2) meters. If a path of uniform width is built around the garden, and the area of path is 40m². Find
a) The expression for the area of path in terms of x
b) Find the value of dimensions of the garden and its area.
If the width of path is 2m, find the area of path.
d) If the garden is measure 12m by 10m, Find the area of path
Solution
Solution:
Let us solve the problem step by step.
Part (a): Expression for the area of the path in terms of
-
The garden has dimensions (length) and (width).
Its area is:
-
If a uniform path is built around the garden with a width , the total dimensions (including the path) become:
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The total area including the path is:
-
The area of the path is the difference between the total area and the garden's area:
Expanding :
Subtracting :
Thus, the area of the path in terms of and is:
Part (b): Find the dimensions and area of the garden
We are told that the area of the path is and the width of the path is :
Substitute into the expression for :
-
Dimensions of the garden:
Length = , Width = . -
Area of the garden:
Part (c): If the width of the path is , find the area of the path.
We already solved this in part (b):
Part (d): If the garden measures , find the area of the path.
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The area of the garden is:
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Total dimensions including the path:
Length = , Width = .Total area including the path:
-
Area of the path is the difference:
Substitute :
Final Answers:
(a) Expression for the path's area:
(b) Dimensions of the garden: , Area = .
(c) Area of the path: .
(d) Area of the path for : .
Follow-up Questions:
- How would the expression for the area of the path change for a different width ?
- What happens to the area of the path if ?
- Can you verify the total area for any and values using the formula?
- How does the area of the path change as the width increases?
- Why does the total area formula include quadratic terms?
Tip:
Always verify your dimensions carefully when dealing with paths around shapes to avoid errors in perimeter or area calculations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Geometry
Area Calculation
Formulas
Area of rectangle = length × width
Expression for total area including path = (length + 2w)(width + 2w)
Area of path = total area - garden area
Theorems
-
Suitable Grade Level
Grades 8-10
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