What are the side lengths of a rectangle if the area is 40 in² and the perimeter is 48 in?

a , b − t h e s i d e l e n g t h s o f a rec t an g l e a ⋅ b = 40 [ i n 2 ] ∧ 2 ⋅ ( a + b ) = 48 [ in ] a + b = 24 ⇒ a = 24 − b ⇒ ab = ( 24 − b ) b = 24 b − b 2 ab = 40 ⇒ 24 b − b 2 = 40 ⇒ − b 2 + 24 b − 40 = 0 / ⋅ ( − 1 ) b 2 − 24 b + 40 = 0 ⇒ Δ = ( − 24 ) 2 − 4 ⋅ 40 = 576 − 160 = 416 = 16 ⋅ 26
Δ ​ = 4 26 ​ ⇒ b 1 ​ = 2 24 − 4 26 ​ ​ = 12 − 2 26 ​ ⇒ a 1 ​ = 12 + 2 26 ​ . b 2 ​ = 2 24 + 4 26 ​ ​ = 12 + 2 26 ​ ⇒ a 2 ​ = 12 − 2 26 ​

Answered by kate200468 | 2024-06-10

The side lengths of the rectangle are approximately 7.3 in and 5.5 in based on the area of 40 in² and the perimeter of 48 in. We used the area and perimeter formulas to derive a quadratic equation that helped us find the side lengths. After solving, we found the two potential dimension pairs from the quadratic results.
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Answered by kate200468 | 2024-12-24