Grade 9 Mathematics: Shadows and Proportionality Calculations

Context and Problem Overview

• Curriculum Reference: This problem is categorized under Grade 9, Unit 5 (G9 U5), identified as Question 4.
• Objective: The task is to determine the height of a specific individual (the girl) based on the proportional relationship between objects and their shadows under identical lighting conditions.
• Scenario: Two individuals, a boy and a girl, are both standing under direct sunlight. This environmental condition ensures that the angle of the sun's rays relative to the ground is the same for both subjects.

Mathematical Principles of Shadows and Proportionality

• Constant Ratio Principle: Under the same sunlight conditions at a specific time, the ratio of an object's height to the length of the shadow it casts is constant. This is because the sun's rays are parallel, forming similar right-angled triangles with the objects and the ground.
• Fundamental Formula:

$\frac{\text{height}}{\text{shadow length}} = \text{constant}$

• This relationship allows for the use of proportions to solve for an unknown variable when other related measurements are known.

Detailed Data and Measurements

• Measurements for the Boy:
  • Height: $1.6\,m$
  • Shadow Length: $2\,m$
• Measurements for the Girl:
  • Height: Unknown (represented by the variable $g$)
  • Shadow Length: $1.8\,m$

Step-by-Step Calculation and Explanation

• Step 1: Establish the Constant Ratio
  • Using the data provided for the boy, calculate the fixed ratio of height to shadow length:

$\frac{1.6}{2} = 0.8$

• Step 2: Formulate the Equation for the Girl
  • Apply the constant ratio ($0.8$) to the girl's unknown height ($g$) and her known shadow length ($1.8\,m$):

$\frac{g}{1.8} = 0.8$

• Step 3: Solve for the Girl's Height
  • To isolate the variable $g$, multiply the shadow length by the constant ratio:

$g = 1.8 \times 0.8$

• Step 4: Final Calculation
  • Perform the multiplication to find the exact value:

$g = 1.44$

Conclusion and Multiple Choice Options

• Analysis of Options: The problem provides four potential answers for the girl's height in meters:
  • $1.68\,m$
  • $1.44\,m$ (Correct)
  • $1.54\,m$
  • $1.5\,m$
• Final Result: Based on the proportional calculation, the height of the girl is exactly $1.44\,m$.