An area model has (z^2+16z+64). What is the area of the small corner?
Answer and explanation
Correct answer: (64)
The expression can be read as a perfect-square identity: \\(z^2+16z+64=(z+8)^2\\). In a visual area model, the large square has side length \\(z+8\\). It can be divided into a \\(z\\)-by-\\(z\\) square, two rectangular parts whose combined area is \\(16z\\), and one small corner square with side length \\(8\\). The corner area is therefore \\(8\times8=64\\).
The constant term represents the area that does not contain \\(z\\). Thus it corresponds to the small fixed square at the corner, giving option A. The term \\(16z\\) represents the combined area of the two rectangles, while \\(z^2\\) represents the large variable square. Therefore, the correct choice is 64, not 16z or \\(z^2\\).
Frequently asked questions
What is the correct answer to this question?
(64)
Why is this the correct answer?
The expression can be read as a perfect-square identity: \\(z^2+16z+64=(z+8)^2\\). In a visual area model, the large square has side length \\(z+8\\). It can be divided into a \\(z\\)-by-\\(z\\) square, two rectangular parts whose combined area is \\(16z\\), and one small corner square with side length \\(8\\). The corner area is therefore \\(8\times8=64\\).
The constant term represents the area that does not contain \\(z\\). Thus it corresponds to the small fixed square at the corner, giving option A. The term \\(16z\\) represents the combined area of the two rectangles, while \\(z^2\\) represents the large variable square. Therefore, the correct choice is 64, not 16z or \\(z^2\\).
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Exploring Algebraic Identities. Topic: Visual models of identities.
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