Question

nti day #1 - area of composite shapes
due: december 12 at 3:00 pm
grade: 50%
area puzzles (rectangles / squares)
write expression with exponents from figure(area)
write an expression involving exponents to represent the shaded area in square feet. then use that expression to calculate how much carpeting deepa will need.
answer attempt 1 out of 2
expressions:
evaluated: square feet

Explanation:

Step1: Find area of large rectangle

Assume the large rectangle has length (let's say) \( l \) and width \( 8 \) ft. Wait, actually, from the diagram (even though not fully clear, typical composite area: shaded area = area of big rectangle - area of unshaded squares. Let's assume big rectangle is, say, length (maybe 8 ft and another side, but wait, the unshaded squares: one is \( 1 \) ft side (so area \( 1^2 \)) and another \( 3 \) ft side (area \( 3^2 \))? Wait, no, maybe the big rectangle is, for example, if the height is 8 ft, and let's say the length is, maybe 8 ft? Wait, no, let's re-express.

Wait, the problem is about composite area: shaded area = area of the large rectangle minus the area of the two unshaded squares (one with side 1 ft, one with side 3 ft). Wait, maybe the large rectangle is a square? Wait, no, let's check the diagram. The large rectangle (shaded) has a height of 8 ft. Let's assume the length is also 8 ft? No, maybe the large rectangle is, say, 8 ft by 8 ft? Wait, no, the unshaded squares: one is 1 ft (area \( 1^2 \)) and one is 3 ft (area \( 3^2 \)). Wait, maybe the large rectangle is 8 ft by 8 ft? Wait, no, let's think again.

Wait, the formula for composite area: shaded area = area of big shape - area of unshaded parts. So if the big shape is a square or rectangle. Let's assume the big rectangle is 8 ft by 8 ft? No, maybe the big rectangle is, for example, 8 ft in height, and let's say the length is, say, 8 ft? Wait, no, maybe the big rectangle is 8 ft by (let's say) 8 ft? Wait, no, the unshaded squares: one with side 1 (area \( 1^2 \)) and one with side 3 (area \( 3^2 \)). So the shaded area would be \( 8 \times 8 - 1^2 - 3^2 \)? Wait, no, maybe the big rectangle is 8 ft by (another length). Wait, maybe the diagram is a square of 8 ft side, with two squares cut out: one 1 ft side, one 3 ft side. So area of big square: \( 8^2 \), area of small square: \( 1^2 \), area of medium square: \( 3^2 \). So shaded area = \( 8^2 - 1^2 - 3^2 \).

So the expression would be \( 8^2 - 1^2 - 3^2 \). Let's compute that: \( 64 - 1 - 9 = 54 \). Wait, but the expression with exponents: \( 8^2 - 1^2 - 3^2 \).

Wait, let's confirm. The large shape is a square with side 8 ft (so area \( 8^2 \)), and we subtract the area of the two unshaded squares: one with side 1 ft (area \( 1^2 \)) and one with side 3 ft (area \( 3^2 \)). So the expression is \( 8^2 - 1^2 - 3^2 \).

Step2: Evaluate the expression

Compute \( 8^2 = 64 \), \( 1^2 = 1 \), \( 3^2 = 9 \). Then \( 64 - 1 - 9 = 54 \). So the expression is \( 8^2 - 1^2 - 3^2 \), and the evaluated area is 54 square feet.

Wait, but maybe the big rectangle is not 8x8. Wait, the diagram shows a vertical side of 8 ft. Maybe the big rectangle is 8 ft by (let's say) 8 ft? Or maybe 8 ft by another length. Wait, maybe the big rectangle is 8 ft in height and 8 ft in length (a square). So area of square: \( 8^2 \), subtract the two squares: \( 1^2 \) and \( 3^2 \). So expression: \( 8^2 - 1^2 - 3^2 \), evaluated as \( 64 - 1 - 9 = 54 \).

Answer:

Expression: \( 8^2 - 1^2 - 3^2 \)
Evaluated: \( 54 \) square feet

(Note: The exact dimensions depend on the diagram, but assuming the large square is 8x8, and unshaded squares 1x1 and 3x3, this is the solution. If the large rectangle is different, adjust accordingly, but based on typical composite area problems, this is a likely scenario.)