Math Problem Statement

the length and breadth of a rectangular metal sheet measured using meter ruler marked in mm are 20.0cm and 12.5cm respectively. calculate the possible error of the area and the percentage possible error of area

Solution

Step 1: Understand the measurements and potential error.

  • Length (LL): 20.0cm20.0 \, \text{cm}
  • Breadth (BB): 12.5cm12.5 \, \text{cm}
  • The measurements are made using a meter ruler marked in millimeters (mm), meaning the smallest unit of measurement is 1mm1 \, \text{mm} or 0.1cm0.1 \, \text{cm}.

The possible error in each measurement is ±0.1cm\pm 0.1 \, \text{cm}.

Step 2: Calculate the area.

The area AA of a rectangle is given by: A=L×BA = L \times B Substitute the given values: A=20.0cm×12.5cm=250.0cm2A = 20.0 \, \text{cm} \times 12.5 \, \text{cm} = 250.0 \, \text{cm}^2

Step 3: Calculate the possible error in the area.

The possible error in the area (ΔA\Delta A) can be estimated using the formula: ΔAA=ΔLL+ΔBB\frac{\Delta A}{A} = \frac{\Delta L}{L} + \frac{\Delta B}{B} Rearranging to find ΔA\Delta A: ΔA=A×(ΔLL+ΔBB)\Delta A = A \times \left( \frac{\Delta L}{L} + \frac{\Delta B}{B} \right) Substitute the given values: ΔLL=0.1cm20.0cm=0.005\frac{\Delta L}{L} = \frac{0.1 \, \text{cm}}{20.0 \, \text{cm}} = 0.005 ΔBB=0.1cm12.5cm=0.008\frac{\Delta B}{B} = \frac{0.1 \, \text{cm}}{12.5 \, \text{cm}} = 0.008 ΔA=250.0cm2×(0.005+0.008)=250.0cm2×0.013=3.25cm2\Delta A = 250.0 \, \text{cm}^2 \times (0.005 + 0.008) = 250.0 \, \text{cm}^2 \times 0.013 = 3.25 \, \text{cm}^2

Step 4: Calculate the percentage error.

The percentage error in the area is given by: Percentage Error=(ΔAA)×100%\text{Percentage Error} = \left( \frac{\Delta A}{A} \right) \times 100\% Substitute the values: Percentage Error=(3.25cm2250.0cm2)×100%=1.3%\text{Percentage Error} = \left( \frac{3.25 \, \text{cm}^2}{250.0 \, \text{cm}^2} \right) \times 100\% = 1.3\%

Final Answers:

  • Possible Error in Area: ±3.25cm2\pm 3.25 \, \text{cm}^2
  • Percentage Error in Area: 1.3%1.3\%

Would you like further details or have any questions?


Here are 5 related questions to explore:

  1. How would the possible error in the area change if the length was 25.0 cm?
  2. What is the possible error in the perimeter of the metal sheet?
  3. How does the percentage error in the area compare to the percentage error in the perimeter?
  4. What happens to the possible error in the area if the ruler's smallest unit of measurement is changed to 0.05 cm?
  5. How would the errors change if the measurements were made in inches instead of centimeters?

Tip: When calculating errors for derived quantities like area or volume, remember that the relative errors of the measurements are additive in such calculations.

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Math Problem Analysis

Mathematical Concepts

Measurement Error
Area Calculation
Percentage Error

Formulas

Area of a Rectangle
Percentage Error Calculation

Theorems

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Suitable Grade Level

High School