Answer :
Let's begin by understanding the given problem step by step with the corresponding calculations.
1. Define Variables:
- Let [tex]\( x \)[/tex] be the side length of the initial square plot of land in meters.
- The initial width of the path is [tex]\( 8 \)[/tex] meters.
2. Area of Initial Gravel Path:
- When you have a square plot of side [tex]\( x \)[/tex] and it's surrounded by a gravel path of width [tex]\( 8 \)[/tex] meters, the outer boundary of the path also forms a square whose side length is [tex]\( x + 2 \times 8 = x + 16 \)[/tex] meters.
- Thus, the area of the larger square (outer boundary) is [tex]\( (x + 16)^2 \)[/tex].
- The area of the initial square plot (inner part) is [tex]\( x^2 \)[/tex].
- Therefore, the area of the initial gravel path is [tex]\( (x + 16)^2 - x^2 \)[/tex].
3. Doubling the Path Width:
- The new width of the path is [tex]\( 16 \)[/tex] meters.
- The new outer boundary of the gravel path forms a square with side length [tex]\( x + 2 \times 16 = x + 32 \)[/tex] meters.
- Thus, the area of the new larger square (outer boundary) is [tex]\( (x + 32)^2 \)[/tex].
- The area of the initial square plot remains [tex]\( x^2 \)[/tex].
- Therefore, the area of the extended gravel path is [tex]\( (x + 32)^2 - x^2 \)[/tex].
4. Given Ratio for the Gravel Required:
- The problem states that the area of the extended gravel path is [tex]\( 1 \frac{2}{9} \)[/tex] or [tex]\( 1 + \frac{2}{9} = \frac{11}{9} \)[/tex] times the area of the initial gravel path.
5. Set Up the Equation:
- Let the area of the initial gravel path be [tex]\( A_{\text{initial}} = (x + 16)^2 - x^2 \)[/tex].
- Let the area of the extended gravel path be [tex]\( A_{\text{extended}} = (x + 32)^2 - x^2 \)[/tex].
- According to the problem, [tex]\( A_{\text{extended}} = \frac{11}{9} A_{\text{initial}} \)[/tex].
6. Solve the Equation:
- Equate the area expressions:
[tex]\[ (x + 32)^2 - x^2 = \frac{11}{9}[(x + 16)^2 - x^2] \][/tex]
7. Expand and Simplify:
- Expand the squares:
[tex]\[ (x + 32)^2 = x^2 + 64x + 1024 \][/tex]
[tex]\[ (x + 16)^2 = x^2 + 32x + 256 \][/tex]
- Substitute to get:
[tex]\[ x^2 + 64x + 1024 - x^2 = \frac{11}{9}(x^2 + 32x + 256 - x^2) \][/tex]
[tex]\[ 64x + 1024 = \frac{11}{9}(32x + 256) \][/tex]
- Multiply to clear the fraction:
[tex]\[ 64x + 1024 = \frac{11}{9} \cdot 32x + \frac{11}{9} \cdot 256 \][/tex]
[tex]\[ 64x + 1024 = \frac{352x}{9} + \frac{2816}{9} \][/tex]
[tex]\[ 9(64x + 1024) = 352x + 2816 \][/tex]
- Simplify further:
[tex]\[ 576x + 9216 = 352x + 2816 \][/tex]
[tex]\[ 576x - 352x = 2816 - 9216 \][/tex]
[tex]\[ 224x = -6400 \][/tex]
[tex]\[ x = \frac{-6400}{224} \][/tex]
[tex]\[ x = -28.57 \][/tex]
8. Conclusion:
- From the equation solving steps, we find the side length [tex]\( x \)[/tex] of the square plot of land to be [tex]\(-28.57\)[/tex] meters. However, a negative result does not make sense in the context of physical dimensions of land. It suggests there might be an error in the problem setup or an unrecognized issue. Reconsidering practical setup, further investigation is warranted, such as reviewing path dimensions or questioning given ratios.
Therefore, while mathematical framework shows how to work through the steps, negative logical results imply deeper review necessity. Practical plot measurements ideally remain non-negative aligning with problem intent and real-world context interpretations.
1. Define Variables:
- Let [tex]\( x \)[/tex] be the side length of the initial square plot of land in meters.
- The initial width of the path is [tex]\( 8 \)[/tex] meters.
2. Area of Initial Gravel Path:
- When you have a square plot of side [tex]\( x \)[/tex] and it's surrounded by a gravel path of width [tex]\( 8 \)[/tex] meters, the outer boundary of the path also forms a square whose side length is [tex]\( x + 2 \times 8 = x + 16 \)[/tex] meters.
- Thus, the area of the larger square (outer boundary) is [tex]\( (x + 16)^2 \)[/tex].
- The area of the initial square plot (inner part) is [tex]\( x^2 \)[/tex].
- Therefore, the area of the initial gravel path is [tex]\( (x + 16)^2 - x^2 \)[/tex].
3. Doubling the Path Width:
- The new width of the path is [tex]\( 16 \)[/tex] meters.
- The new outer boundary of the gravel path forms a square with side length [tex]\( x + 2 \times 16 = x + 32 \)[/tex] meters.
- Thus, the area of the new larger square (outer boundary) is [tex]\( (x + 32)^2 \)[/tex].
- The area of the initial square plot remains [tex]\( x^2 \)[/tex].
- Therefore, the area of the extended gravel path is [tex]\( (x + 32)^2 - x^2 \)[/tex].
4. Given Ratio for the Gravel Required:
- The problem states that the area of the extended gravel path is [tex]\( 1 \frac{2}{9} \)[/tex] or [tex]\( 1 + \frac{2}{9} = \frac{11}{9} \)[/tex] times the area of the initial gravel path.
5. Set Up the Equation:
- Let the area of the initial gravel path be [tex]\( A_{\text{initial}} = (x + 16)^2 - x^2 \)[/tex].
- Let the area of the extended gravel path be [tex]\( A_{\text{extended}} = (x + 32)^2 - x^2 \)[/tex].
- According to the problem, [tex]\( A_{\text{extended}} = \frac{11}{9} A_{\text{initial}} \)[/tex].
6. Solve the Equation:
- Equate the area expressions:
[tex]\[ (x + 32)^2 - x^2 = \frac{11}{9}[(x + 16)^2 - x^2] \][/tex]
7. Expand and Simplify:
- Expand the squares:
[tex]\[ (x + 32)^2 = x^2 + 64x + 1024 \][/tex]
[tex]\[ (x + 16)^2 = x^2 + 32x + 256 \][/tex]
- Substitute to get:
[tex]\[ x^2 + 64x + 1024 - x^2 = \frac{11}{9}(x^2 + 32x + 256 - x^2) \][/tex]
[tex]\[ 64x + 1024 = \frac{11}{9}(32x + 256) \][/tex]
- Multiply to clear the fraction:
[tex]\[ 64x + 1024 = \frac{11}{9} \cdot 32x + \frac{11}{9} \cdot 256 \][/tex]
[tex]\[ 64x + 1024 = \frac{352x}{9} + \frac{2816}{9} \][/tex]
[tex]\[ 9(64x + 1024) = 352x + 2816 \][/tex]
- Simplify further:
[tex]\[ 576x + 9216 = 352x + 2816 \][/tex]
[tex]\[ 576x - 352x = 2816 - 9216 \][/tex]
[tex]\[ 224x = -6400 \][/tex]
[tex]\[ x = \frac{-6400}{224} \][/tex]
[tex]\[ x = -28.57 \][/tex]
8. Conclusion:
- From the equation solving steps, we find the side length [tex]\( x \)[/tex] of the square plot of land to be [tex]\(-28.57\)[/tex] meters. However, a negative result does not make sense in the context of physical dimensions of land. It suggests there might be an error in the problem setup or an unrecognized issue. Reconsidering practical setup, further investigation is warranted, such as reviewing path dimensions or questioning given ratios.
Therefore, while mathematical framework shows how to work through the steps, negative logical results imply deeper review necessity. Practical plot measurements ideally remain non-negative aligning with problem intent and real-world context interpretations.
