Question

Дана правильная треугольная пирамида, сторона основания которой 12 дм. найдите объём пирамиды, если её боковой ребро образует с плоскостью основания угол 45 градусов

Answer 1

Подробное решение во вложениях. Если что не понятно пиши в лс.

Удачи.

Answer 2

Calculation of the Volume of a Triangular Pyramid

To find the volume of a triangular pyramid, we need to know the length of the base and the height of the pyramid. In this case, we are given that the base of the pyramid has a side length of 12 dm and that the angle between the lateral edge and the base is 45 degrees.

To calculate the volume, we can use the formula:

Volume = (1/3) * base area * height

The base area can be calculated using the formula for the area of an equilateral triangle:

Base area = (sqrt(3)/4) * side length^2

Let's calculate the volume step by step.

1. Calculate the base area: - The side length of the base is given as 12 dm. - Using the formula for the area of an equilateral triangle, we can calculate the base area: Base area = (sqrt(3)/4) * 12^2.

2. Calculate the height of the pyramid: - The angle between the lateral edge and the base is given as 45 degrees. - To find the height, we can use trigonometry. - The height can be calculated as the product of the base length and the sine of the angle: Height = side length * sin(angle).

3. Calculate the volume: - Now that we have the base area and the height, we can calculate the volume using the formula: Volume = (1/3) * base area * height.

Let's perform the calculations.

1. Calculate the base area: - Base area = (sqrt(3)/4) * 12^2 - Base area = (1.732/4) * 144 - Base area = 0.433 * 144 - Base area = 62.352 dm^2.

2. Calculate the height of the pyramid: - Height = side length * sin(angle) - Height = 12 * sin(45 degrees) - Height = 12 * 0.707 - Height = 8.484 dm.

3. Calculate the volume: - Volume = (1/3) * base area * height - Volume = (1/3) * 62.352 * 8.484 - Volume ≈ 167.918 dm^3.

Therefore, the volume of the given triangular pyramid is approximately 167.918 cubic decimeters (dm^3).