Answer:
460 cm
Step-by-step explanation:
Shapes F and G are two different shapes that each have an area of 3² × 23² cm². Shape F is a square, while Shape G is a rectangle with integer side lengths.
The total area of each shape is 3² × 23² = 4761 cm².
The area of a rectangle is the product of its width and length. Therefore, to find the possible side lengths of rectangle G, we need to find the factor pairs of its area, 4761 cm².
The prime factorization of 4761 is already given as 3² × 23², so the prime factors are 3 and 23.
For each prime factor, we can have 0 or 1 or 2 of the factor, because it's raised to the power of 2 in the prime factorization. So, we can compute the factors of 4761 as follows:
3⁰ × 23⁰ = 1
3¹ × 23⁰ = 3
3² × 23⁰ = 9
3⁰ × 23¹ = 23
3¹ × 23¹ = 69
3² × 23¹ = 207
3⁰ × 23² = 529
3¹ × 23² = 1587
3² × 23² = 4761
Therefore, the factors of 4761 are 1, 3, 9, 23, 69, 207, 529, 1587, and 4761.
As the area of a rectangle is the product of its width and length, and the area of rectangle G is 4761 cm², the width and length of rectangle G have to be factor pairs of 4761, which are:
1 and 4761
3 and 1587
9 and 529
23 and 207
69 and 69
The perimeter of a rectangle is twice the sum of its width and length. So, the corresponding perimeters for the factor pairs are:
1 and 4761 → Perimeter = 2(1 + 4761) = 9524 cm
3 and 1587 → Perimeter = 2(3 + 1587) = 3180 cm
9 and 529 → Perimeter = 2(9 + 529) = 1076 cm
23 and 207 → Perimeter = 2(23 + 207) = 460 cm
69 and 69 → Perimeter = 2(69 + 69) = 276 cm
Therefore, the smallest possible perimeter for a shape with an area of 4761 cm², where its side lengths are integers, is 276 cm. However, this perimeter corresponds to a square due to the equal width and length.
Since the question specifies that shapes F and G are different shapes with the same area, and that shape F is a square, the perimeter of 276 cm must be attributed to shape F, as it represents that of a square.
Therefore, we consider the next smallest perimeter for rectangle G, which is:
[tex]\LARGE\boxed{\boxed{460 \; \sf cm}}[/tex]
Additional Notes
A square is a special type of rectangle where all four sides are equal in length. However, not all rectangles have equal side lengths, so a rectangle is not necessarily a square.
