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Solve for Dimensions of a Rectangular Field Split into Two Squares with 224 Yards of Fencing
Mathematics
Grade 8 of Junior High School
Question Content
A rectangular field is divided into 2 squares of the same size and shape. If it takes 224 yards of fencing to enclose the field and divide the field into the two parcels, find the dimensions of the field. See the figure. The width, W, of the field is ____
Correct Answer
Width = 32 yards, Length = 64 yards
Detailed Solution Steps
1
Step 1: Define variables. Let the width of the rectangular field (which is equal to the side length of each square) be $W$. Since the field is made of two identical squares placed side by side, the length of the rectangle is $2W$.
2
Step 2: Count the total fencing segments. The fencing includes the outer perimeter of the rectangle plus the dividing fence down the middle. This gives a total of 5 segments of length $W$ (2 widths for the sides, 2 lengths split into 2 $W$ segments each, plus 1 dividing segment).
3
Step 3: Set up the equation for total fencing. The total fencing is 224 yards, so $5W = 224$.
4
Step 4: Solve for $W$. Divide both sides of the equation by 5: $W = 224 \\div 5 = 32$ yards.
5
Step 5: Find the length. The length is $2W = 2 \\times 32 = 64$ yards.
Knowledge Points Involved
1
Rectangular and Square Dimensions Relationship
When a rectangle is formed by two identical squares, one side of the rectangle is equal to the side length of the square, and the other side is twice that length. This relationship is used to link the variables for length and width of the rectangle.
2
Perimeter and Fencing Calculation
For problems involving fencing that includes internal dividers, total fencing is the sum of the outer perimeter and all internal dividing segments, not just the standard rectangular perimeter formula $P=2(l+w)$.
3
Linear Equation with One Variable
A linear equation of the form $ax = b$ (where $a$ and $b$ are constants) is used to solve for an unknown variable. It is solved by isolating the variable through division or multiplication, which is the core method for solving basic algebraic word problems.
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