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Find the value of y: Square and Rectangle with Equal Areas
Mathematics
Grade 7 (Junior High School)
Question Content
This square and rectangle have the same area. Work out the value of y. The square has side length 6 cm, the rectangle has height 4 cm and base length y cm. (Not drawn accurately)
Correct Answer
9 cm
Detailed Solution Steps
1
Step 1: Calculate the area of the square. The formula for the area of a square is Area = side × side. Substitute the side length 6 cm: Area of square = 6 × 6 = 36 cm².
2
Step 2: Use the fact that the area of the rectangle equals the area of the square (given in the problem), so the area of the rectangle is also 36 cm².
3
Step 3: Use the formula for the area of a rectangle: Area = length × width. We know the width (height) is 4 cm and the area is 36 cm², so set up the equation: 4 × y = 36.
4
Step 4: Solve for y by dividing both sides of the equation by 4: y = 36 ÷ 4 = 9.
Knowledge Points Involved
1
Area of a Square
The area of a square is calculated by multiplying the length of one side by itself, expressed as the formula \(A = s^2\), where \(s\) is the length of a side. This applies to all squares, as all four sides are equal in length, and it measures the 2-dimensional space enclosed by the square's edges.
2
Area of a Rectangle
The area of a rectangle is calculated by multiplying its length by its width, expressed as the formula \(A = l \times w\), where \(l\) is the length (longer side) and \(w\) is the width (shorter side). It measures the 2-dimensional space enclosed by the rectangle's four right-angle edges.
3
Equating Areas to Solve for Unknowns
When two shapes are stated to have equal areas, we can set their area formulas equal to each other and solve for an unknown dimension. This is a common algebraic application in geometry, used to find missing side lengths when area relationships are given.
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