Question

боковое ребро правильной треугольной пирамиды равно 6 см составляет с плоскостью основания 60 градусов. Найти объем пирамиды.   1) H=6*sin60=3√3 отрезок высоты = 2/3*h=6*cos60=6*1/2=3 h=9/2 a=h/sin60=9/2 /1/2= 9 S=a^2√3/4=9^2√3/4=81√3/4 V=1/3*S*h=1/3*81√3/4 *9/2=243√3/8   2) 6*sin60=6*sqrt(3)/2=3*sqrt(3) S=3R^2sqrt(3)/4 R=6*cos60=3 S=3*9sqrt(3)/4=27sqrt(3)/4 V=1/3hS=27*sqrt(3)*3sqrt(3)/3*4=81/4=20,25 какой из решений правильный?

Answer 1

Calculation of the Volume of a Triangular Pyramid

To find the volume of a triangular pyramid, we need to know the length of one of its lateral edges and the angle it forms with the base plane. In this case, we are given that the length of the lateral edge is 6 cm and it forms a 60-degree angle with the base plane.

There are two different approaches to calculating the volume of the pyramid, and we will explore both of them.

Approach 1: Using Height and Base Area

1. First, let's find the height of the pyramid. We can use the formula `h = a * sin(60)`, where `a` is the length of the lateral edge. Substituting the given value, we have `h = 6 * sin(60) = 6 * √3 / 2 = 3√3` cm. 2. Next, we can find the area of the base. The base of the pyramid is an equilateral triangle, so its area can be calculated using the formula `S = a^2 * √3 / 4`, where `a` is the length of one side of the triangle. Substituting the value of `a` as `h / sin(60)`, we have `S = (h / sin(60))^2 * √3 / 4 = (9/2)^2 * √3 / 4 = 81√3 / 4` cm². 3. Finally, we can calculate the volume of the pyramid using the formula `V = (1/3) * S * h`. Substituting the values we have calculated, we get `V = (1/3) * (81√3 / 4) * (3√3) = 243√3 / 8` cm³.

Approach 2: Using Slant Height and Base Area

1. The slant height of the pyramid can be calculated using the formula `l = a * sin(60)`, where `a` is the length of the lateral edge. Substituting the given value, we have `l = 6 * sin(60) = 6 * √3 / 2 = 3√3` cm. 2. Next, we can find the area of the base using the formula `S = 3R^2 * √3 / 4`, where `R` is the circumradius of the base triangle. The circumradius can be calculated using the formula `R = a * cos(60)`, where `a` is the length of the lateral edge. Substituting the given value, we have `R = 6 * cos(60) = 6 * 1/2 = 3` cm. Substituting the value of `R` into the formula for `S`, we get `S = 3 * 3^2 * √3 / 4 = 27√3 / 4` cm². 3. Finally, we can calculate the volume of the pyramid using the formula `V = (1/3) * h * S`. Substituting the values we have calculated, we get `V = (1/3) * 3√3 * (27√3 / 4) = 81 / 4 = 20.25` cm³.

Both approaches yield different results for the volume of the pyramid. According to the calculations:

- Approach 1 gives a volume of `243√3 / 8` cm³. - Approach 2 gives a volume of `20.25` cm³.

It is important to note that the correct approach depends on the specific context and assumptions made in the problem. Without further information, it is difficult to determine which approach is correct.