Question
Write an inequality to represent the following situation. Then, find the solution.
An architect designs a window in the shape of a rectangle. The clients have purchased 30 feet of trim to frame the window. If the window needs to have an area of at least 20 square feet to meet city requirements, what are the possible lengths of the window?
An architect designs a window in the shape of a rectangle. The clients have purchased 30 feet of trim to frame the window. If the window needs to have an area of at least 20 square feet to meet city requirements, what are the possible lengths of the window?
Answers
Answered by
GPT-4o mini
To represent the situation with inequalities, let's define the variables:
- Let \( l \) be the length of the rectangle (window).
- Let \( w \) be the width of the rectangle (window).
### Step 1: Perimeter Constraint
The clients have purchased 30 feet of trim for the framing of the window. The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2l + 2w
\]
Since they have 30 feet of trim, we can write the inequality:
\[
2l + 2w \leq 30
\]
Dividing the entire inequality by 2 gives:
\[
l + w \leq 15
\]
### Step 2: Area Requirement
The area \( A \) of the rectangle is given by:
\[
A = l \times w
\]
The requirement states that the area must be at least 20 square feet, so we write another inequality:
\[
l \times w \geq 20
\]
### Step 3: Solving for Width
Now, we can express \( w \) in terms of \( l \) using the perimeter inequality:
\[
w \leq 15 - l
\]
Substituting this expression for \( w \) into the area inequality:
\[
l \times (15 - l) \geq 20
\]
### Step 4: Expanding and Rearranging
Expanding the left side gives:
\[
15l - l^2 \geq 20
\]
Rearranging the inequality results in:
\[
-l^2 + 15l - 20 \geq 0
\]
Multiplying through by -1 (and remembering to flip the inequality sign):
\[
l^2 - 15l + 20 \leq 0
\]
### Step 5: Finding Roots of the Quadratic
To solve \( l^2 - 15l + 20 = 0 \), we apply the quadratic formula:
\[
l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 1, b = -15, c = 20 \):
\[
l = \frac{15 \pm \sqrt{(-15)^2 - 4 \times 1 \times 20}}{2 \times 1}
\]
\[
l = \frac{15 \pm \sqrt{225 - 80}}{2}
\]
\[
l = \frac{15 \pm \sqrt{145}}{2}
\]
Calculating \( \sqrt{145} \approx 12.04\):
\[
l = \frac{15 \pm 12.04}{2}
\]
Calculating both possible values gives:
\[
l_1 = \frac{27.04}{2} \approx 13.52
\]
\[
l_2 = \frac{2.96}{2} \approx 1.48
\]
### Step 6: Interval for \( l \)
The quadratic \( l^2 - 15l + 20 \) opens upwards (since the coefficient of \( l^2 \) is positive) and is less than or equal to zero between its two roots. Therefore, the lengths must satisfy:
\[
1.48 \leq l \leq 13.52
\]
### Conclusion
The possible lengths \( l \) of the window that satisfy both the area requirement and the trim limitation are given by the interval:
\[
\boxed{[1.48, 13.52]}
\]
- Let \( l \) be the length of the rectangle (window).
- Let \( w \) be the width of the rectangle (window).
### Step 1: Perimeter Constraint
The clients have purchased 30 feet of trim for the framing of the window. The perimeter \( P \) of a rectangle is given by the formula:
\[
P = 2l + 2w
\]
Since they have 30 feet of trim, we can write the inequality:
\[
2l + 2w \leq 30
\]
Dividing the entire inequality by 2 gives:
\[
l + w \leq 15
\]
### Step 2: Area Requirement
The area \( A \) of the rectangle is given by:
\[
A = l \times w
\]
The requirement states that the area must be at least 20 square feet, so we write another inequality:
\[
l \times w \geq 20
\]
### Step 3: Solving for Width
Now, we can express \( w \) in terms of \( l \) using the perimeter inequality:
\[
w \leq 15 - l
\]
Substituting this expression for \( w \) into the area inequality:
\[
l \times (15 - l) \geq 20
\]
### Step 4: Expanding and Rearranging
Expanding the left side gives:
\[
15l - l^2 \geq 20
\]
Rearranging the inequality results in:
\[
-l^2 + 15l - 20 \geq 0
\]
Multiplying through by -1 (and remembering to flip the inequality sign):
\[
l^2 - 15l + 20 \leq 0
\]
### Step 5: Finding Roots of the Quadratic
To solve \( l^2 - 15l + 20 = 0 \), we apply the quadratic formula:
\[
l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 1, b = -15, c = 20 \):
\[
l = \frac{15 \pm \sqrt{(-15)^2 - 4 \times 1 \times 20}}{2 \times 1}
\]
\[
l = \frac{15 \pm \sqrt{225 - 80}}{2}
\]
\[
l = \frac{15 \pm \sqrt{145}}{2}
\]
Calculating \( \sqrt{145} \approx 12.04\):
\[
l = \frac{15 \pm 12.04}{2}
\]
Calculating both possible values gives:
\[
l_1 = \frac{27.04}{2} \approx 13.52
\]
\[
l_2 = \frac{2.96}{2} \approx 1.48
\]
### Step 6: Interval for \( l \)
The quadratic \( l^2 - 15l + 20 \) opens upwards (since the coefficient of \( l^2 \) is positive) and is less than or equal to zero between its two roots. Therefore, the lengths must satisfy:
\[
1.48 \leq l \leq 13.52
\]
### Conclusion
The possible lengths \( l \) of the window that satisfy both the area requirement and the trim limitation are given by the interval:
\[
\boxed{[1.48, 13.52]}
\]
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