Question

Дана правильная треугольная пирамида, объем которой равен 432. Боковое ребро этой пирамиды наклонено к плоскости основания под углом 45⁰. Найдите радиус основания конуса, описанного около данной пирамиды.

Answer 1

Ответ:

$R=4\sqrt[6]{243}$

Объяснение:

Основание правильной пирамиды - правильный треугольник. Пусть а - его сторона.

Радиус основания конуса, описанного около пирамиды - радиус окружности, описанной около правильного треугольника.

$R=OA=\dfrac{a\sqrt{3}}{3}=\dfrac{a}{\sqrt{3}}$

$a=R\sqrt{3}$

ΔSAO:  ∠SOA = 90°,  ∠SAO = 45°,  ⇒ ∠ASO = 45°, треугольник равнобедренный.

h = R.

Объем пирамиды:

$V=\dfrac{1}{3}S_{ABC}\cdot h$

$S_{ABC}=\dfrac{a^2\sqrt{3}}{4}$

$V=\dfrac{1}{3}\dfrac{a^2\sqrt{3}}{4}\cdot h=\dfrac{a^2\sqrt{3}\cdot h}{12}$

Учитывая, что

$a=R\sqrt{3}$  и  $h=R$, получим:

$V=\dfrac{(R\sqrt{3})^2\cdot \sqrt{3}\cdot R}{12}=\dfrac{R^2\cdot 3\cdot \sqrt{3}\cdot R}{12}=\dfrac{R^3\sqrt{3}}{4}=432$

$R^3=\dfrac{432\cdot 4}{\sqrt{3}}=\dfrac{432\cdot 4\cdot \sqrt{3}}{3}=576\sqrt{3}$

$R=\sqrt[3]{576\sqrt{3}}=4\sqrt[3]{9\sqrt{3}}=4\sqrt[3]{\sqrt{243}}$

$R=4\sqrt[6]{243}$

Возможно, в условии допущена ошибка: объем пирамиды равен 432√3. Тогда вычисления в конце выглядят проще:

$\dfrac{R^3\sqrt{3}}{4}=432\sqrt{3}$

$R^3=432\cdot 4=1728$

$R=\sqrt[3]{1728}=12$

Answer 2

Problem Analysis

We are given a right triangular pyramid with a volume of 432. One of the lateral edges of the pyramid is inclined to the base plane at an angle of 45 degrees. We need to find the radius of the base of the cone circumscribed around the given pyramid.

Solution

To solve this problem, we can use the formula for the volume of a pyramid and the relationship between the volume of a pyramid and the volume of a cone.

Let's denote the radius of the base of the cone as r and the height of the pyramid as h.

The volume of a pyramid is given by the formula: V = (1/3) * base_area * height.

The volume of a cone is given by the formula: V_cone = (1/3) * π * r^2 * h_cone.

Since the pyramid and the cone are circumscribed around each other, their volumes are equal: V = V_cone.

Substituting the formulas for the volumes, we get: (1/3) * base_area * height = (1/3) * π * r^2 * h_cone.

Since the pyramid is right triangular, the base area can be calculated using the formula for the area of a right triangle: base_area = (1/2) * base * height_base.

We know that one of the lateral edges of the pyramid is inclined to the base plane at an angle of 45 degrees. This means that the height of the pyramid is equal to the base of the triangle.

Substituting the formulas for the base area and the height, we get: (1/3) * (1/2) * base * height_base * height_base = (1/3) * π * r^2 * h_cone.

Simplifying the equation, we have: (1/6) * base * height_base^2 = (1/3) * π * r^2 * h_cone.

Since the height of the pyramid is equal to the base of the triangle, we can rewrite the equation as: (1/6) * base * base^2 = (1/3) * π * r^2 * h_cone.

Simplifying further, we get: (1/6) * base^3 = (1/3) * π * r^2 * h_cone.

Since the base of the pyramid is a right triangle, we can use the Pythagorean theorem to find the length of the base: base = √(height_base^2 + height_base^2) = √(2 * height_base^2) = √2 * height_base.

Substituting this into the equation, we have: (1/6) * (√2 * height_base)^3 = (1/3) * π * r^2 * h_cone.

Simplifying further, we get: (1/6) * 2^(3/2) * height_base^3 = (1/3) * π * r^2 * h_cone.

Since the height of the pyramid is equal to the base of the triangle, we can rewrite the equation as: (1/6) * 2^(3/2) * height^3 = (1/3) * π * r^2 * h_cone.

We are given that the volume of the pyramid is 432, so we can substitute this value into the equation: (1/6) * 2^(3/2) * height^3 = (1/3) * π * r^2 * h_cone = 432.

Now we can solve for the radius of the base of the cone, r.

Calculation

Let's calculate the radius of the base of the cone using the given information.

We are given that the volume of the pyramid is 432, so we can substitute this value into the equation: (1/6) * 2^(3/2) * height^3 = (1/3) * π * r^2 * h_cone = 432.

Simplifying the equation, we have: (1/6) * 2^(3/2) * height^3 = 144 * π * r^2.

Dividing both sides of the equation by 144 * π, we get: (1/6) * 2^(3/2) * height^3 / (144 * π) = r^2.

Taking the square root of both sides of the equation, we have: r = √((1/6) * 2^(3/2) * height^3 / (144 * π)).

Substituting the value of height = base = h, we get: r = √((1/6) * 2^(3/2) * h^3 / (144 * π)).

Since the height of the pyramid is inclined to the base plane at an angle of 45 degrees, we can use the relationship between the height and the lateral edge of the pyramid: h = lateral_edge / √2.

Substituting this into the equation, we have: r = √((1/6) * 2^(3/2) * (lateral_edge / √2)^3 / (144 * π)).

Simplifying further, we get: r = √((1/6) * 2^(3/2) * lateral_edge^3 / (144 * π * 2^(3/2))).

Simplifying the expression, we have: r = √(lateral_edge^3 / (432 * π)).

Now we can substitute the given value of the volume of the pyramid, V = 432, and the given angle of inclination, angle = 45 degrees, to find the value of the lateral edge.

Using the formula for the volume of a pyramid, we have: V = (1/3) * base_area * height.

Substituting the formula for the base area of a right triangle, we have: 432 = (1/3) * (1/2) * base * height.

Since the height of the pyramid is equal to the base of the triangle, we can rewrite the equation as: 432 = (1/3) * (1/2) * base * base.

Simplifying the equation, we have: 432 = (1/6) * base^2.

Multiplying both sides of the equation by 6, we get: 2592 = base^2.

Taking the square root of both sides of the equation, we have: base = √2592.

Since the base of the pyramid is a right triangle, the base can be expressed as base = a * √2, where a is the length of one of the legs of the right triangle.

Substituting this into the equation, we have: a * √2 = √2592.

Simplifying the equation, we have: a = √(2592 / 2) = √1296 = 36.

Since the lateral edge is inclined to the base plane at an angle of 45 degrees, we can use the relationship between the lateral edge and the base of the right triangle: lateral_edge = base * √2.

Substituting the values of base = 36 and angle = 45 degrees, we get: lateral_edge = 36 * √2.

Now we can substitute the value of the lateral edge into the equation for the radius of the base of the cone: r = √(lateral_edge^3 / (432 * π)) = √((36 * √2)^3 / (432 * π)).

Calculating the value of r, we get: r ≈ 5.03.

Answer

The radius of the base of the cone circumscribed around the given pyramid is approximately 5.03.