Study Notes on Circumference, Area, Volume & Surface Area Errors

Trigonometric Function: Cosine

  • The function given is ( \cos(4x) ).

Circumference of a Circle

  • Measurement: The circumference of a circle is measured at 64 centimeters.
  • Possible Error: There is a possible error in this measurement of 0.9 centimeters.

Problems

Problem 32: Circumference and Area Errors

  1. (a) Approximate the percent error in computing the area of the circle.

    • Formula for Circumference:( C = 2\pi r )
      • Implying: ( r = \frac{C}{2\pi} = \frac{64}{2\pi} )
    • Derive the Area:
      • Area ( A = \pi r^2 = \pi \left( \frac{64}{2\pi} \right)^2 = \pi \left( \frac{32^2}{\pi^2} \right) = \frac{1024}{\pi} )
    • Calculating the area error:
      • If circumference error is 0.9 cm, compute error in radius:
      • ( r + \Delta r = \frac{(64 + 0.9)}{2\pi} )
      • Thus, calculate new area ( A + \Delta A ).
    • Percent Error Formula:
      • ( ext{Percent Error} = \frac{|\Delta A|}{A} \times 100 \%

  2. (b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 3%.

    • Given: Maximum area percent error (= 3\%)
    • ( \Delta A = \frac{3}{100} A )\
    • Determine associated circumference error using derivation of area concerning circumference:
      • Establish relationship between ( C ) and ( A )
      • From area formula, during differentiation:
      • ( \Delta A = \frac{\partial A}{\partial C} \Delta C )
      • Set ( \Delta A ) to maximum limit and solve for ( \Delta C ).
Problem 33: Volume and Surface Area
  1. Volume and Surface Area of Cube:
    • Measurement: The edge length of the cube is measured as 15 inches.
    • Possible Errors: Include variations in this measurement to calculate volume and surface area errors.
    • Volume Calculation:
      • Volume ( V = s^3 = 15^3 )
      • ( V = 3375 \text{ cubic inches} )
    • Surface Area Calculation:
      • Surface Area ( A = 6s^2 = 6(15^2) = 6(225) )
      • ( A = 1350 \text{ square inches} )
    • Range of Error Calculation:
      • Acknowledge potential inaccuracies in measuring ( s ).
    • Use same error percentage principles as in Problem 32 to derive acceptable discrepancies in volume and surface area measurements.

Conclusion

  • The calculations and estimations derived offer crucial insights into the impacts of measuring errors on geometric properties. Students must grasp the relationship among different properties and their sensitivities to measurement inaccuracies.