Question

Найти объем тела вращения полученного при вращении равнобедренного треугольника с катетом 6 см вокруг его оси симметрии

Answer 1

Вращением треугольника вокруг своей оси симметрии получаем конус.
Длина его образующей равна 6см. По теореме Пифагора найдем квадрат диаметра:
$d^2 = 6^2 + 6^2 = 72$
Значит:
$d = 6 \sqrt{2}$
Радиус основания конуса равен:
$r = 0.5d = 3 \sqrt{2}$

Площадь основания равна:
$\pi R^2 = 18 \pi$

Найдём высоту конуса:
$h^2 = 6^2 - (3 \sqrt{2})^2 = 36 - 18 = 18$
$h = 3 \sqrt{2}$

Объем конуса равен:
$V = \frac{1}{3} S_{o}h = \frac{1}{3} 18 \pi 3 \sqrt{2} = 18 \sqrt{2} \pi$

Answer 2

Calculation of the Volume of the Rotated Triangle

To find the volume of the solid obtained by rotating an isosceles triangle with a leg length of 6 cm around its axis of symmetry, we can use the formula for the volume of a solid of revolution.

The formula for the volume of a solid of revolution is given by:

V = πR^2h

Where: - V is the volume of the solid - π is a mathematical constant approximately equal to 3.14159 - R is the distance from the axis of rotation to the outer edge of the solid (in this case, the distance from the axis of symmetry to the base of the triangle) - h is the height of the solid (in this case, the length of the leg of the triangle)

In this case, the triangle is isosceles, so the distance from the axis of symmetry to the base of the triangle is equal to half the length of the base. Therefore, R = 6/2 = 3 cm.

To find the height of the solid, we can use the Pythagorean theorem. The height of the triangle can be found by calculating the length of the hypotenuse using the leg length of 6 cm.

Using the Pythagorean theorem, we have:

h = √(leg^2 + base^2)

Substituting the values, we get:

h = √(6^2 + 6^2) = √(36 + 36) = √72 ≈ 8.485 cm

Now we can substitute the values of R and h into the formula for the volume:

V = π(3^2)(8.485) ≈ 254.47 cm^3

Therefore, the volume of the solid obtained by rotating the isosceles triangle with a leg length of 6 cm around its axis of symmetry is approximately 254.47 cm^3.

Answer

The volume of the solid obtained by rotating the isosceles triangle with a leg length of 6 cm around its axis of symmetry is approximately 254.47 cm^3.