Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.

Verified step by step guidance

1. Start by recalling the formulas for the perimeter and area of a rectangle. The perimeter \( P \) is given by \( P = 2l + 2w \), where \( l \) is the length and \( w \) is the width. The area \( A \) is given by \( A = l \times w \).
2. Substitute the given perimeter into the perimeter formula: \( 36 = 2l + 2w \). Simplify this equation to express one variable in terms of the other, for example, \( l = 18 - w \).
3. Substitute the expression for \( l \) from the perimeter equation into the area formula: \( 77 = (18 - w) \times w \).
4. Expand the equation \( 77 = 18w - w^2 \) and rearrange it to form a quadratic equation: \( w^2 - 18w + 77 = 0 \).
5. Solve the quadratic equation \( w^2 - 18w + 77 = 0 \) using the quadratic formula \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -18 \), and \( c = 77 \). This will give you the possible values for \( w \), and you can find \( l \) using \( l = 18 - w \).

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the rectangle, calculated by the formula P = 2(length + width). In this problem, the perimeter is given as 36 feet, which provides a relationship between the length and width of the rectangle that can be used to set up equations.

Area of a Rectangle

The area of a rectangle is the amount of space enclosed within its sides, calculated using the formula A = length × width. In this case, the area is specified as 77 square feet, which allows for another equation that relates the length and width, enabling the solution of the problem.

System of Equations

A system of equations consists of two or more equations that share variables. In this scenario, the two equations derived from the perimeter and area can be solved simultaneously to find the values of length and width. Techniques such as substitution or elimination can be employed to solve the system.