are measurement

Area Measurement and Carpet Calculation

• Raldo measured the dimensions of a room:

  • Length = 9 feet
  • Width = 10 feet
  • Calculated area: 90 square feet
  • Formula: Area = Length × Width = 9 ft × 10 ft = 90 sq ft

• Raldo intends to carpet the room:

  • Carpet measurement: square yards
  • Ordered amount of carpet: 30 square yards

Incorrect Calculation of Carpet Amount

• Assessment of Carpeting Needs:
  • Raldo's conversion approach:
  • Believed to get square yards, he incorrectly calculated as: 90 sq ft / 3 = 30 sq yards
  • Misinterpretation:
  • Did not account for squaring the dimensions in area measurement; conversion is in one-dimensional units, not two-dimensional:
    • Square feet (sq ft) vs. square yards (sq yds)
    • Conversion error: Required to assess square measurements instead of linear

Correct Conversion of Square Feet to Square Yards

• To convert the area of the room from square feet to square yards:
  • Start with the area:
  • 90 square feet = 90 square feet ( = 90 ext{ ft}^2 = 90 ext{ ft} imes ext{ ft})
  • Recognize the need for a unit fraction to convert:
  • Conversion factor: 1 yard = 3 feet
  • Conversion steps:
  • Write 90 square feet as a fraction: 90 square feet/1
  • Establish unit fraction:
    • ( rac{1 ext{ yard}}{3 ext{ feet}} )
  • Since area involves squaring the dimensions, apply the unit fraction twice:
    • ( rac{1 ext{ yard}}{3 ext{ feet}} imes rac{1 ext{ yard}}{3 ext{ feet}} )
• Multiplying through:
  • Result of multiplication:
  • Numerator: 90
  • Denominator: 3 × 3 = 9
  • Resulting units: square yards ( ext{yd}^2)
• Final Calculation:
  • Final conversion computation:
  • 90 ÷ 9 = 10 square yards
  • Conclusion: Raldo should have ordered 10 square yards of carpet, not 30.

Visualizing the Area Conversion

• Raldo's calculation rationale:

  • Understanding square yard area:
  • Visualize one square yard:
  • When cut into square feet:
    • Vertical cuts: 3 equal parts
    • Horizontal cuts: 3 equal parts
  • Total count of square feet in one square yard: 9 square feet

• Conclusion of this section:

  • Corrected understanding leads to the conclusion that 90 square feet equals 10 square yards. Raldo should order 10 square yards.

Estimation of Brick Patio Costs

• Cost of a smaller brick patio:
  • Dimensions:
  • 15 feet by 20 feet
  • Area: ( 15 ext{ ft} imes 20 ext{ ft} = 300 ext{ ft}^2 )
  • Installation cost: $2,275

Calculating Cost of a Larger Patio

• Dimensions of larger patio:
  • 21 feet by 25 feet
  • Area Calculation:
  • Area = 21 feet × 25 feet = 525 square feet
• Estimating cost using proportions:
  • Define variables:
  • Area of small patio: ( A_1 = 300 ext{ ft}^2 )
  • Area of large patio: ( A_2 = 525 ext{ ft}^2 )
  • Cost of small patio: ( C_1 = 2275 )
  • Unknown cost of large patio: ( C_2 )
  • Establish proportion:
  • ( \frac{C1}{A1} = \frac{C2}{A2} )
• Cross multiplication to solve for ( C_2 ):
  • Equation:
  • 300C_2 = 2,275 × 525
  • Calculate right side: 2,275 × 525 = 1,194,375
  • So, 300C_2 = 1,194,375
  • Solve for C_2:
  • ( C_2 = \frac{1,194,375}{300} = 3,981.25 )

Rounding the Estimate

• Instruction to round:
  • Round to the nearest dollar.
  • Rounding Process:
  • 3,981.25 rounds down to $3,981 (due to tenths place being 2).
• Conclusion of this section:
  • Estimated cost for the larger patio is $3,981.

Alternative Calculation Strategy

• Alternative method for cost estimation:
  • Find cost per square foot of the small patio:
  • Cost per square foot = ( \frac{C1}{A1} = \frac{2275}{300} )
  • Multiply by area of larger patio:
  • Cost = Cost per square foot × Area of larger patio = cost per square foot × 525 square feet.
  • Confirming results yields the same estimate as earlier.

Conclusion

• Summary of findings:
  • Raldo's carpet needs corrected from 30 to 10 square yards.
  • Cost estimation methods agree on the projected cost for the patio installation, illustrating the viability of different solving approaches.