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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?

They are the simplest shapes and perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes 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?

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

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

Area Signup and view all the answers

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

$4 \times \text{side length}$ Signup and view all the answers

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

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

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

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

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

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