Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![## Question #22: Find Height and Length of Rectangle
### Given Information:
- **Area (A)**: \( 60 \text{ in}^2 \)
- **Length (L)**: \( 3x + 3 \)
- **Height (H)**: \( x + 2 \)
### Diagram:
A rectangle is drawn with the area labeled as \( A = 60 \text{ in}^2 \). On the diagram:
- The length (L) is labeled as \( 3x + 3 \) along one of the longer sides.
- The height (H) is labeled as \( x + 2 \) along one of the shorter sides.
### Objective:
To find the values of \( x \) for the given length and height of the rectangle and subsequently determine the numerical values of the length and height.
### Process:
1. **Using the area formula for a rectangle**:
\[ A = L \times H \]
2. **Substitute the given values**:
\[ 60 = (3x + 3)(x + 2) \]
3. **Solve for \( x \)**:
- Expand and simplify the equation:
\[ 60 = 3x^2 + 6x + 3x + 6 \]
\[ 60 = 3x^2 + 9x + 6 \]
\[ 0 = 3x^2 + 9x - 54 \] (Subtract 60 from both sides)
- Simplify by dividing the entire equation by 3:
\[ 0 = x^2 + 3x - 18 \]
- Factor the quadratic equation:
\[ 0 = (x + 6)(x - 3) \]
- Solve for \( x \):
\[ x = -6 \quad \text{or} \quad x = 3 \]
4. **Determine feasible value (positive)**:
\[ x = 3 \]
5. **Calculate Length (L) and Height (H)**:
- \( L = 3x + 3 \)
\[ L = 3(3) + 3 \]
\[ L = 9 + 3 \]
\[ L = 12 \text{ in} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b46bad1-a760-4ca0-8910-9e1779c10984%2F6e0bcda2-a7df-4e9a-b8c4-af8c721e7cf6%2Fufbbet_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Question #22: Find Height and Length of Rectangle
### Given Information:
- **Area (A)**: \( 60 \text{ in}^2 \)
- **Length (L)**: \( 3x + 3 \)
- **Height (H)**: \( x + 2 \)
### Diagram:
A rectangle is drawn with the area labeled as \( A = 60 \text{ in}^2 \). On the diagram:
- The length (L) is labeled as \( 3x + 3 \) along one of the longer sides.
- The height (H) is labeled as \( x + 2 \) along one of the shorter sides.
### Objective:
To find the values of \( x \) for the given length and height of the rectangle and subsequently determine the numerical values of the length and height.
### Process:
1. **Using the area formula for a rectangle**:
\[ A = L \times H \]
2. **Substitute the given values**:
\[ 60 = (3x + 3)(x + 2) \]
3. **Solve for \( x \)**:
- Expand and simplify the equation:
\[ 60 = 3x^2 + 6x + 3x + 6 \]
\[ 60 = 3x^2 + 9x + 6 \]
\[ 0 = 3x^2 + 9x - 54 \] (Subtract 60 from both sides)
- Simplify by dividing the entire equation by 3:
\[ 0 = x^2 + 3x - 18 \]
- Factor the quadratic equation:
\[ 0 = (x + 6)(x - 3) \]
- Solve for \( x \):
\[ x = -6 \quad \text{or} \quad x = 3 \]
4. **Determine feasible value (positive)**:
\[ x = 3 \]
5. **Calculate Length (L) and Height (H)**:
- \( L = 3x + 3 \)
\[ L = 3(3) + 3 \]
\[ L = 9 + 3 \]
\[ L = 12 \text{ in} \]
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