In the rectangle below, the line MN cuts the rectangle into two regions. Find x the length of segment NB so that the area of the quadrilateral MNBC is 40% of the tota area of the rectangle. A 10 m D 5 m M 20 m NX B C
In the rectangle below, the line MN cuts the rectangle into two regions. Find x the length of segment NB so that the area of the quadrilateral MNBC is 40% of the tota area of the rectangle. A 10 m D 5 m M 20 m NX B C
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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![### Problem Statement:
In the rectangle below, the line MN cuts the rectangle into two regions. Find \( x \), the length of segment NB, so that the area of the quadrilateral MNBC is 40% of the total area of the rectangle.
#### Diagram Description:
- A rectangle labeled \( ABCD \) where:
- \( A \) is the top-left corner.
- \( B \) is the top-right corner.
- \( C \) is the bottom-right corner.
- \( D \) is the bottom-left corner.
- The rectangle has the following dimensions:
- Length \( AB = 20 \) meters.
- Height \( AD = 10 \) meters.
- Point \( M \) is located on line segment \( DC \), 5 meters from \( D \).
- Point \( N \) is located on line segment \( AB \), \( x \) meters from \( B \).
- Line \( MN \) intersects the rectangle, forming quadrilateral \( MNBC \).
### Solution Steps:
1. **Calculate the total area of the rectangle:**
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 20 \text{ m} \times 10 \text{ m} = 200 \text{ m}^2
\]
2. **Calculate 40% of the total area of the rectangle:**
\[
0.40 \times \text{Area}_{\text{rectangle}} = 0.40 \times 200 \text{ m}^2 = 80 \text{ m}^2
\]
3. **Find \( x \) such that the area of quadrilateral \( MNBC \) equals 80 \( \text{m}^2 \):**
- Divide the complex shape into simpler shapes or use coordinate geometry to find the areas.
- Utilize the properties of the rectangle and line segments to establish relationships.
Since specialized calculations may involve multi-step geometric analysis for an exact \( x \), an educational approach can involve guided exercises in proportions and geometric principles.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57a7d70-87de-4a1f-8104-5b2578062c6c%2F1f19b6f3-ccda-4532-b27a-9e668a0139d1%2Fcs8vibc_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
In the rectangle below, the line MN cuts the rectangle into two regions. Find \( x \), the length of segment NB, so that the area of the quadrilateral MNBC is 40% of the total area of the rectangle.
#### Diagram Description:
- A rectangle labeled \( ABCD \) where:
- \( A \) is the top-left corner.
- \( B \) is the top-right corner.
- \( C \) is the bottom-right corner.
- \( D \) is the bottom-left corner.
- The rectangle has the following dimensions:
- Length \( AB = 20 \) meters.
- Height \( AD = 10 \) meters.
- Point \( M \) is located on line segment \( DC \), 5 meters from \( D \).
- Point \( N \) is located on line segment \( AB \), \( x \) meters from \( B \).
- Line \( MN \) intersects the rectangle, forming quadrilateral \( MNBC \).
### Solution Steps:
1. **Calculate the total area of the rectangle:**
\[
\text{Area}_{\text{rectangle}} = \text{Length} \times \text{Width} = 20 \text{ m} \times 10 \text{ m} = 200 \text{ m}^2
\]
2. **Calculate 40% of the total area of the rectangle:**
\[
0.40 \times \text{Area}_{\text{rectangle}} = 0.40 \times 200 \text{ m}^2 = 80 \text{ m}^2
\]
3. **Find \( x \) such that the area of quadrilateral \( MNBC \) equals 80 \( \text{m}^2 \):**
- Divide the complex shape into simpler shapes or use coordinate geometry to find the areas.
- Utilize the properties of the rectangle and line segments to establish relationships.
Since specialized calculations may involve multi-step geometric analysis for an exact \( x \), an educational approach can involve guided exercises in proportions and geometric principles.
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