Calculating Rectangular Plot Gravel Path Width

The problem involves finding the uniform width of a gravel path inside a rectangular grassy plot given the plot dimensions and the path's area.

1. Initial Information

• Outer Rectangle (Grassy Plot) Length ($L$): 38 m
• Outer Rectangle Width ($W$): 32 m
• Area of the Gravel Path ($A_{path}$): $600 \text{ m}^2$
• Let the uniform width of the path be $x$ meters.

2. Dimensions of the Inner Rectangle

The path is inside the plot, so the inner dimensions (the grassy area excluding the path) are reduced by $2x$ (width $x$ on both sides).

• Inner Length ($l$): $L - 2x = 38 - 2x$ m
• Inner Width ($w$): $W - 2x = 32 - 2x$ m

For the inner dimensions to be valid, $38 - 2x > 0$ and $32 - 2x > 0$, which means $2x < 32$, or $x < 16$ m.

3. Area Calculation

The area of the outer rectangle ($A_{outer}$) is $L \times W$. The area of the inner rectangle ($A_{inner}$) is $l \times w$. The area of the path is the difference between these two areas.

Outer Area:

Inner Area:

Path Area Equation:

4. Solving the Equation

Substitute the expression for $A_{inner}$:

Rearrange into a standard quadratic equation ($ax^2 + bx + c = 0$):

Divide the entire equation by 4 to simplify:

Factor the quadratic equation:

This gives two possible values for $x$:

5. Final Answer Determination

We must consider the physical constraint that the width of the path cannot be larger than half of the plot's width. As determined earlier, $x < 16$ m.

• $x = 30$ m is not a valid solution because it is greater than 16 m.
• $x = 5$ m is a valid solution as it is less than 16 m.

Therefore, the width of the path is 5 m.