Higher Order Thinking Isabella has three rectangular cards that are 4 inches by 5 inches. How can she arrange the cards, without overlapping, to make one larger polygon with the smallest possible perimeter? How will the area of the polygon compare to the combined area of the three cards?

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Calculate Area: Question_prompt: What is the arrangement of three 4-inch by 5-inch rectangular cards that results in the smallest possible perimeter for the combined shape, and how does the area of this shape compare to the combined area of the three individual cards?Calculate Combined Area: First, let's calculate the area of one card. The area of a rectangle is found by multiplying its length by its width. Area of one card = length × width Area of one card = 4 inches × 5 inches Area of one card = 20 square inchesConsider Arrangements: Next, we calculate the combined area of the three cards by multiplying the area of one card by three. Combined area = Area of one card × 3 Combined area = 20 square inches × 3 Combined area = 60 square inchesRow Arrangement: Now, let's consider the possible arrangements of the cards. To minimize the perimeter, we want to minimize the exposed edges. The best way to do this is to arrange the cards so that they share as many edges as possible.L-Shaped Arrangement: One way to arrange the cards is to place them side by side in a row. This would create a new rectangle that is 4 inches by 15 inches. However, this may not be the arrangement with the smallest perimeter.Calculate Perimeter (Row): Another way is to arrange the cards so that two cards are placed side by side and the third card is placed below them, creating an L-shape. This would result in a shape that is 8 inches by 9 inches.Calculate Perimeter (L-Shaped): To determine which arrangement has the smallest perimeter, we calculate the perimeter for each arrangement. The perimeter of a rectangle is found by adding up all its sides. Perimeter of the row arrangement = 2 × (length + width) Perimeter of the row arrangement = 2 × (4 inches + 15 inches) Perimeter of the row arrangement = 2 × 19 inches Perimeter of the row arrangement = 38 inchesCorrect Perimeter Calculation: Now, we calculate the perimeter of the L-shaped arrangement. Perimeter of the L-shaped arrangement = 2 × (length + width) + length Perimeter of the L-shaped arrangement = 2 × (8 inches + 9 inches) + 4 inches Perimeter of the L-shaped arrangement = 2 × 17 inches + 4 inches Perimeter of the L-shaped arrangement = 34 inches + 4 inches Perimeter of the L-shaped arrangement = 38 inchesWe made a mistake in the previous step. The L-shaped arrangement does not have a perimeter that can be calculated using the formula for a rectangle because it is not a rectangle. We need to calculate the perimeter by adding the individual sides. Perimeter of the L-shaped arrangement = 8 inches + 9 inches + 8 inches + 9 inches + 4 inches + 5 inches Perimeter of the L-shaped arrangement = 43 inches

Higher Order Thinking Isabella has three rectangular cards that are $4$ inches by $5$ inches. How can she arrange the cards, without overlapping, to make one larger polygon with the smallest possible perimeter? How will the area of the polygon compare to the combined area of the three cards?

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Higher Order Thinking Isabella has three rectangular cards that are $4$ inches by $5$ inches. How can she arrange the cards, without overlapping, to make one larger polygon with the smallest possible perimeter? How will the area of the polygon compare to the combined area of the three cards?