Geometry and Area Calculation Study Notes

Introduction

• Discussion on a complex problem involving geometry and area calculations.
• The scenario involves calculating an unknown length in an outdoor eating space at a school.

Problem Overview

• Diagram Reference: On page 547, there is a diagram representing a school’s outdoor eating space.
• Area Relation: The area of the outdoor eating space is given as 750 square yards.

Objective

• The primary objective is to find the unknown length represented as x in the context of the problem.

Methodology

Decomposition of Shapes

• Decomposing the Shape: Since the shape is irregular, it is beneficial to decompose it into simpler geometric figures:
  • Rectangles
  • Triangles

Identifying Components

• Labeling Segments: The sections of the decomposed shape are labeled as follows:
  • a: First rectangle
  • b: Second rectangle
  • c: Triangle

Area Calculation for Component a

• Formula for Area of a Rectangle: Area = Length × Width
• Known Dimensions:
  • Length = 16 yards
  • Width = 27 yards
• Calculation:
  • Area of a = $16 imes 27$
  • Area of a = $432$ square yards

Area Calculation for Component b

• Given: Length = 12 yards; Width = x yards
• Calculation:
  • Area of b = $12 imes x$
  • Area of b = $12x$ square yards

Area Calculation for Component c

• Formula for Area of a Triangle: Area = ½ × Base × Height
• Known Dimensions:
  • Height = 12 yards
  • Base = $32 - x$ yards
• Calculation:
  • Area of c = $\frac{1}{2} imes 12 imes (32 - x)$
  • Area of c = $6(32 - x)$
  • Area of c = $192 - 6x$ square yards

Total Area Calculation

• Summing All Areas: The total area (A_total) can be defined as:
  • $A_{total} = A_a + A_b + A_c$
• Substituting calculated areas:
  • $A_{total} = 432 + 12x + (192 - 6x)$
• Simplifying the expression:
  • $A_{total} = 432 + 192 + 12x - 6x$
  • $A_{total} = 624 + 6x$

Setting Up the Equation

• Equating Total Area to Given Area: The problem states the total area is 750 square yards. Thus:
  • $624 + 6x = 750$

Solving for x

1. Isolate x: Subtract 624 from both sides:
   • $6x = 750 - 624$
   • $6x = 126$
2. Dividing: To find x, divide by 6:
   • $x = \frac{126}{6}$
   • $x = 21$ yards

Conclusion

• Value of x: The unknown length x has been determined to be 21 yards.
• Ensures that the calculations align with the conditions and components defined in the problem statement, appropriately utilizing geometric principles to arrive at the solution.