Perimeters and Quiz grade 6
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Questions and Answers

What is the formula for computing the length of a closed piecewise smooth plane curve $\gamma : [a, b] \to \mathbb{R}^2$?

  • $L = \int_{a}^{b} \sqrt{x'(t)^{2} + y'(t)^{2}} dt$ (correct)
  • $L = \int_{a}^{b} (x'(t) - y'(t)) dt$
  • $L = \int_{a}^{b} (x'(t) + y'(t)) dt$
  • $L = \int_{a}^{b} (x'(t) \cdot y'(t)) dt$
  • Who is the mathematician known to have approximated the perimeter of a circle by surrounding it with regular polygons?

  • Archimedes (correct)
  • Pythagoras
  • Euclid
  • Newton
  • Which theory describes a generalized notion of perimeter, including hypersurfaces bounding volumes in n-dimensional Euclidean spaces?

  • Riemann integration
  • Caccioppoli sets (correct)
  • Complex analysis
  • Fourier series
  • Why are polygons fundamental to determining perimeters?

    <p>They are the simplest shapes and perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes</p> Signup and view all the answers

    In the formula for computing the length of a closed piecewise smooth plane curve, what does $x'(t)$ represent?

    <p>The derivative of the x-coordinate function with respect to t</p> Signup and view all the answers

    What is the measurement of space enclosed by a closed geometric figure?

    <p>Area</p> Signup and view all the answers

    Which formula is used to calculate the total length of the boundary of a square field?

    <p>$4 \times \text{side length}$</p> Signup and view all the answers

    What is the formula for calculating the area of a triangle?

    <p>$\frac{1}{2} \times \text{base} \times \text{height}$</p> Signup and view all the answers

    What is the formula for calculating the area of a circle?

    <p>$\pi \times \text{radius}^2$</p> Signup and view all the answers

    What is the formula for calculating the perimeter of a rectangle?

    <p>$2 \times (\text{length} + \text{width})$</p> Signup and view all the answers

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