Question

Основание равнобедренной трапеции равны 11 и 21см,а боковая сторона равна 13см. Вычислить объем фигуры,образуемой при вращении этой тропеции вокруг ее оси

Answer 1

При вращении равнобедренной трапеции вокруг оси симметрии получится усеченный конус.
r=5.5 cм,
R=10.5 cм. Высоту можно вычислить по теореме Пифагора: h= √(13²-5²) = 12 см.
V= 1/3 *π* h*(r²+R²+R*r) = 1/3 *π*12*(30.25+110.25+57.75)=4π*198.25=
793π cм³

Answer 2

Calculation of the Volume of the Figure Formed by Rotating an Isosceles Trapezoid

To calculate the volume of the figure formed by rotating an isosceles trapezoid around its axis, we need to determine the shape of the resulting solid.

Let's consider the given isosceles trapezoid with base lengths of 11 cm and 21 cm, and a side length of 13 cm.

To visualize the resulting solid, imagine rotating the trapezoid around its axis. This rotation will create a three-dimensional figure known as a frustum of a cone. The frustum of a cone is a solid with a circular base, a smaller circular top, and a curved surface connecting the two bases.

To calculate the volume of the frustum of a cone, we need to know the radii of the two circular bases and the height of the frustum.

In this case, the larger base of the frustum corresponds to the longer base of the trapezoid, which is 21 cm. The smaller base of the frustum corresponds to the shorter base of the trapezoid, which is 11 cm. The height of the frustum is equal to the side length of the trapezoid, which is 13 cm.

Now, let's calculate the radii of the two circular bases of the frustum. The radius of the larger base is half the length of the longer base, so it is 21 cm / 2 = 10.5 cm. The radius of the smaller base is half the length of the shorter base, so it is 11 cm / 2 = 5.5 cm.

Using the formula for the volume of a frustum of a cone, which is V = (1/3) * π * h * (R^2 + r^2 + R * r), where V is the volume, π is a mathematical constant approximately equal to 3.14159, h is the height of the frustum, R is the radius of the larger base, and r is the radius of the smaller base, we can calculate the volume of the figure.

Substituting the values into the formula, we have:

V = (1/3) * 3.14159 * 13 cm * (10.5 cm^2 + 5.5 cm^2 + 10.5 cm * 5.5 cm)

Calculating this expression, we find:

V ≈ 750 cm^3

Therefore, the volume of the figure formed by rotating the given isosceles trapezoid around its axis is approximately 750 cubic centimeters.