Question

1. Основою піраміди є рівнобедре- ний трикутник із основою 12 см і кутом 120° при вершині. Усі бічні ребра утворюють із площи- ною основи кут 30°. Знайдіть ви- соту піраміди.ДАЮ 100 БАЛОВ,НУЖНО С УСЛОВИЕМ ЗАДАЧИ

Answer 1

Finding the Volume of the Pyramid

To find the volume of the given pyramid, we can use the formula for the volume of a pyramid, which is given by the formula:

Volume = (1/3) * Base Area * Height

In this case, we are given that the base of the pyramid is an equilateral triangle with a side length of 12 cm and the angle at the apex is 120°. The lateral edges form a 30° angle with the base.

Calculating the Base Area

The base of the pyramid is an equilateral triangle. The formula to find the area of an equilateral triangle is:

Area = (sqrt(3) / 4) * side^2

Substituting the given side length of 12 cm into the formula, we get:

Area = (sqrt(3) / 4) * 12^2

Calculating the Height of the Pyramid

To find the height of the pyramid, we can use trigonometric relationships. Since the lateral edges form a 30° angle with the base, we can use the trigonometric function tangent to find the height.

tan(30°) = Opposite / Adjacent

The opposite side is the height of the pyramid, and the adjacent side is half the base of the equilateral triangle.

Calculation

Using the given information, we can calculate the volume of the pyramid as follows:

Area = (sqrt(3) / 4) * 12^2 Height = 12 * tan(30°)

Substituting the values: Area = (sqrt(3) / 4) * 144 Height = 12 * tan(30°)

Now, we can use the formula for the volume of a pyramid to find the volume.

Volume = (1/3) * (sqrt(3) / 4) * 144 * (12 * tan(30°))

After calculating the above expression, we get the volume of the pyramid.

I hope this helps! If you have further questions or need additional assistance, feel free to ask.