To determine the possible lengths of a rectangular garden with a given perimeter and a minimum area requirement, follow these steps:
1. Understand the problem:
- The perimeter (P) of the garden is 150 feet.
- The area (A) of the garden must be at least 1200 square feet.
- You need to find the possible lengths (L) of the garden.
2. Formulate the perimeter equation:
- The perimeter of a rectangle is given by [tex]\( P = 2L + 2W \)[/tex], where L is the length and W is the width.
- Given [tex]\( P = 150 \)[/tex], this equation becomes:
[tex]\[
2L + 2W = 150
\][/tex]
- Simplify to:
[tex]\[
L + W = 75
\][/tex]
- From this, you can express W in terms of L:
[tex]\[
W = 75 - L
\][/tex]
3. Formulate the area inequality:
- The area of a rectangle is given by [tex]\( A = L \times W \)[/tex].
- Given [tex]\( A \geq 1200 \)[/tex]:
[tex]\[
L \times W \geq 1200
\][/tex]
- Substitute [tex]\( W \)[/tex] with [tex]\( 75 - L \)[/tex]:
[tex]\[
L \times (75 - L) \geq 1200
\][/tex]
4. Set up the quadratic inequality:
- Expand the equation:
[tex]\[
L \times (75 - L) \geq 1200
\][/tex]
[tex]\[
75L - L^2 \geq 1200
\][/tex]
- Rearrange to form a standard quadratic inequality:
[tex]\[
L^2 - 75L + 1200 \leq 0
\][/tex]
5. Solve the quadratic equation:
- Use the quadratic formula [tex]\( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = -75 \)[/tex], and [tex]\( c = 1200 \)[/tex].
- Solving the quadratic equation yields two critical points, which are the roots of the equation:
[tex]\[
L_1 = 23 \quad \text{and} \quad L_2 = 52
\][/tex]
6. Interpret the results:
- These roots represent the boundary values for the length of the garden.
- To satisfy both the perimeter and area conditions, the possible lengths of the garden must be within the range defined by these roots.
Therefore, the possible lengths of the rectangular garden are between 23 feet and 52 feet, inclusive.
