Difficulty: Easy
Correct Answer: 24 cm
Explanation:
Introduction / Context:
This question tests the relationship between the perimeter and area of squares and how to work with differences of areas. By using basic formulas for perimeter and area, you can move from perimeters to side lengths, then to areas, and finally to the required new square whose area equals the difference of the first two areas.
Given Data / Assumptions:
Concept / Approach:
First find the side lengths of the two given squares using the perimeter formula. Next, compute their areas. Subtract the smaller area from the larger area to find the area of the new square. Then take the square root to get the new side length and multiply by four to obtain the perimeter of the third square.
Step-by-Step Solution:
First square: P1 = 40 cm, so side s1 = 40 / 4 = 10 cm
Area A1 = s1^2 = 10^2 = 100 cm^2
Second square: P2 = 32 cm, so side s2 = 32 / 4 = 8 cm
Area A2 = s2^2 = 8^2 = 64 cm^2
Area of third square A3 = A1 - A2 = 100 - 64 = 36 cm^2
Side of third square s3 = sqrt(36) = 6 cm
Perimeter of third square = 4 * s3 = 4 * 6 = 24 cm
Verification / Alternative check:
If the third square has side 6 cm, its area is 36 cm^2. Adding this to the second square area 64 cm^2 gives 100 cm^2, which matches the first square area. This confirms that we have correctly interpreted the difference and found the right perimeter.
Why Other Options Are Wrong:
12 cm and 20 cm: Correspond to smaller side lengths that do not give area 36 cm^2.
36 cm and 48 cm: Correspond to side lengths 9 cm and 12 cm, with areas much larger than 36 cm^2.
Common Pitfalls:
Subtracting perimeters instead of areas.
Forgetting that perimeter uses side linearly while area uses side squared.
Taking square root incorrectly when finding the side of the third square.
Final Answer:
The perimeter of the third square is 24 cm
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