Question

Основание прямой призмы служит равнобедренный треугольник с углом 120 градусов при вершине . Диагональ грани , противоположной этому углу, имеет длину 6 см и составляет с плоскостью основания угол 60 градусов . Найти объём призмы

Answer 1

Всё подробно написала в решении.

Answer 2

Finding the Base of the Prism

To find the base of the prism, we are given that it is an isosceles triangle with an angle of 120 degrees at the vertex. Additionally, we know that the diagonal of the face opposite this angle has a length of 6 cm and forms a 60-degree angle with the base plane.

Let's break down the problem step by step:

1. We have an isosceles triangle with an angle of 120 degrees at the vertex. This means that the other two angles of the triangle are equal. Let's call this triangle ABC, with angle BAC measuring 120 degrees.

2. The diagonal of the face opposite the 120-degree angle has a length of 6 cm and forms a 60-degree angle with the base plane. Let's call this diagonal AD, with angle DAB measuring 60 degrees.

3. To find the base of the prism, we need to find the length of the equal sides of the isosceles triangle ABC.

4. Using trigonometry, we can find the length of the equal sides of the triangle. Since angle BAC is 120 degrees and angle DAB is 60 degrees, we can use the sine rule to find the length of side AB.

The sine rule states that for any triangle ABC, the ratio of the length of a side to the sine of the opposite angle is constant. In this case, we have:

AB / sin(120 degrees) = AD / sin(60 degrees)

Rearranging the equation, we get:

AB = (AD * sin(120 degrees)) / sin(60 degrees)

Plugging in the values, we have:

AB = (6 cm * sin(120 degrees)) / sin(60 degrees)

Evaluating the trigonometric functions, we find:

AB ≈ 6 cm * 0.866 / 0.866 ≈ 6 cm

Therefore, the length of side AB is approximately 6 cm.

5. Since the triangle ABC is isosceles, the length of side BC is also 6 cm.

Finding the Volume of the Prism

Now that we know the base of the prism is an isosceles triangle with side lengths of 6 cm, we can find the volume of the prism.

The volume of a prism is given by the formula:

Volume = Base Area * Height

1. The base area of the prism is the area of the isosceles triangle ABC. To find this, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, the base of the triangle is 6 cm and the height can be found using the Pythagorean theorem. Let's call the height h.

Using the Pythagorean theorem, we have:

h^2 = AB^2 - (BC/2)^2

Plugging in the values, we get:

h^2 = 6^2 - (6/2)^2 = 36 - 9 = 27

Taking the square root of both sides, we find:

h ≈ √27 ≈ 5.196 cm

Therefore, the height of the triangle (and the prism) is approximately 5.196 cm.

Now we can calculate the area of the triangle:

Area = (1/2) * 6 cm * 5.196 cm ≈ 15.588 cm^2

2. The height of the prism is the same as the height of the triangle, which is approximately 5.196 cm.

3. Now we can calculate the volume of the prism using the formula:

Volume = Base Area * Height

Plugging in the values, we have:

Volume ≈ 15.588 cm^2 * 5.196 cm ≈ 80.925 cm^3

Therefore, the volume of the prism is approximately 80.925 cubic centimeters.

Conclusion

In conclusion, the base of the prism is an isosceles triangle with side lengths of 6 cm, and the volume of the prism is approximately 80.925 cubic centimeters.