Math Problem Statement

C. Case Based Question

  1. 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 xx

  1. The garden has dimensions xx (length) and x2x - 2 (width).
    Its area is:
    Agarden=x(x2)=x22xA_{\text{garden}} = x(x - 2) = x^2 - 2x

  2. If a uniform path is built around the garden with a width ww, the total dimensions (including the path) become:
    Length: x+2w,Width: (x2)+2w=x2+2w\text{Length: } x + 2w, \quad \text{Width: } (x - 2) + 2w = x - 2 + 2w

  3. The total area including the path is:
    Atotal=(x+2w)(x2+2w)A_{\text{total}} = (x + 2w)(x - 2 + 2w)

  4. The area of the path is the difference between the total area and the garden's area:
    Apath=AtotalAgardenA_{\text{path}} = A_{\text{total}} - A_{\text{garden}}

Expanding AtotalA_{\text{total}}:
Atotal=(x+2w)(x2+2w)=x2+2wx2x+2wx4w+4w2A_{\text{total}} = (x + 2w)(x - 2 + 2w) = x^2 + 2wx - 2x + 2wx - 4w + 4w^2 Atotal=x2+4wx2x4w+4w2A_{\text{total}} = x^2 + 4wx - 2x - 4w + 4w^2

Subtracting Agarden=x22xA_{\text{garden}} = x^2 - 2x:
Apath=(x2+4wx2x4w+4w2)(x22x)A_{\text{path}} = \big(x^2 + 4wx - 2x - 4w + 4w^2\big) - \big(x^2 - 2x\big) Apath=4wx4w+4w2A_{\text{path}} = 4wx - 4w + 4w^2

Thus, the area of the path in terms of xx and ww is:
Apath=4w(x1)+4w2A_{\text{path}} = 4w(x - 1) + 4w^2


Part (b): Find the dimensions and area of the garden

We are told that the area of the path is 40m240 \, \text{m}^2 and the width of the path is 2m2 \, \text{m}:
Substitute w=2w = 2 into the expression for ApathA_{\text{path}}:
Apath=4(2)(x1)+4(22)A_{\text{path}} = 4(2)(x - 1) + 4(2^2) 40=8(x1)+1640 = 8(x - 1) + 16 40=8x8+1640 = 8x - 8 + 16 40=8x+840 = 8x + 8 8x=328x = 32 x=4x = 4

  1. Dimensions of the garden:
    Length = x=4mx = 4 \, \text{m}, Width = x2=42=2mx - 2 = 4 - 2 = 2 \, \text{m}.

  2. Area of the garden:
    Agarden=x(x2)=4(2)=8m2.A_{\text{garden}} = x(x - 2) = 4(2) = 8 \, \text{m}^2.


Part (c): If the width of the path is 2m2 \, \text{m}, find the area of the path.

We already solved this in part (b):
Apath=40m2A_{\text{path}} = 40 \, \text{m}^2


Part (d): If the garden measures 12m×10m12 \, \text{m} \times 10 \, \text{m}, find the area of the path.

  1. The area of the garden is:
    Agarden=12×10=120m2A_{\text{garden}} = 12 \times 10 = 120 \, \text{m}^2

  2. Total dimensions including the path:
    Length = 12+2w12 + 2w, Width = 10+2w10 + 2w.

    Total area including the path:
    Atotal=(12+2w)(10+2w)A_{\text{total}} = (12 + 2w)(10 + 2w)

  3. Area of the path is the difference:
    Apath=AtotalAgardenA_{\text{path}} = A_{\text{total}} - A_{\text{garden}} Substitute w=2w = 2:
    Atotal=(12+4)(10+4)=16×14=224m2A_{\text{total}} = (12 + 4)(10 + 4) = 16 \times 14 = 224 \, \text{m}^2 Apath=224120=104m2A_{\text{path}} = 224 - 120 = 104 \, \text{m}^2


Final Answers:

(a) Expression for the path's area:
Apath=4w(x1)+4w2A_{\text{path}} = 4w(x - 1) + 4w^2

(b) Dimensions of the garden: 4m×2m4 \, \text{m} \times 2 \, \text{m}, Area = 8m28 \, \text{m}^2.

(c) Area of the path: 40m240 \, \text{m}^2.

(d) Area of the path for 12m×10m12 \, \text{m} \times 10 \, \text{m}: 104m2104 \, \text{m}^2.


Follow-up Questions:

  1. How would the expression for the area of the path change for a different width ww?
  2. What happens to the area of the path if w=3mw = 3 \, \text{m}?
  3. Can you verify the total area for any xx and ww values using the formula?
  4. How does the area of the path change as the width ww increases?
  5. 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

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Suitable Grade Level

Grades 8-10