Final answer:
To find the dimensions of a rectangle with an area of 30 cm^2 and a perimeter of 26 cm, we set up two equations: L × W = 30 and 2(L + W) = 26. Solving these equations, we determine that the length and width of the rectangle are either 10 cm by 3 cm or 3 cm by 10 cm.
Step-by-step explanation:
The area of a rectangle is found using the formula Area = length × width, and the perimeter is found using the formula Perimeter = 2 × (length + width). Given that the area is 30 cm2 and the perimeter is 26 cm, we can set up two equations with two variables to represent the length (L) and width (W) of the rectangle:
- L × W = 30
- 2 × (L + W) = 26
We can simplify the perimeter equation to get L + W = 13. Next, we can express one variable in terms of the other using the area equation, for instance, W = 30/L. Now, substituting this expression into the simplified perimeter equation gives us L + (30/L) = 13.
We can solve this equation for the length by multiplying through by L to get rid of the fraction: L2 + 30 = 13L. Rearranging the terms, we have L2 - 13L + 30 = 0. Factoring the quadratic equation, we find that (L - 10)(L - 3) = 0, which gives us two possible lengths: L = 10 cm or L = 3 cm. Since L × W = 30, if L is 10 cm, then W must be 3 cm, and vice versa. Therefore, the dimensions of the rectangle are a length of 10 cm and a width of 3 cm, or a length of 3 cm and a width of 10 cm.