Final answer:
For the rectangle with a perimeter of 44 and sides as consecutive even integers, the length is 12. For the problem with three consecutive integers, the middle integer is 11. Lastly, for the triangle with sides related by a factor of 4 and a perimeter of 43 cm, the smallest side is 5 cm.
Step-by-step explanation:
Let's tackle each problem one by one. For the first problem, concerning the rectangle's perimeter, we'll use consecutive even integers to represent the length and the width. If we let x be the width, then the length will be x + 2 since they are consecutive even numbers. The formula for perimeter (P) is P = 2(length + width). Plugging in the values and solving for x gives us:
44 = 2(x + (x + 2)) → 44 = 2(2x + 2) → 44 = 4x + 4 → 4x = 40 → x = 10.
Since x represents width, the length is x + 2 = 12. Therefore, the length of the rectangle is 12.
For the second problem involving consecutive integers, let y be the first integer, then the second will be y + 1, and the third will be y + 2. The equation based on the given information becomes:
(y + (y + 1)) - (y + 2) = 9 → 2y + 1 - y - 2 = 9 → y = 10.
The middle integer, therefore, is y + 1 = 11.
Lastly, for the triangle with a perimeter of 43 cm, let the smallest side be z. Then the largest side is 4z and the other side is 2z + 8. The perimeter equation becomes:
43 = z + 4z + (2z + 8) → 43 = 7z + 8 → 7z = 35 → z = 5.
Thus, the length of the smallest side of the triangle is 5 cm.