Question

Спасите!!!!! Расстояние между любыми двумя боковыми ребрами наклонной треугольной призмы равно 2. Боковое ребро равно 3 и наклонено к плоскости основания под углом 30°. Найдите площадь полной поверхности призмы.

Answer 1

Площадь S‍1 ‍ боковой поверхности призмы равна произведению периметра перпендикулярного сечения призмы на её боковое ребро. Плоскость перпендикулярного сечения пересекает боковые грани по их высотам. Поэтому периметр перпендикулярного сечения равен сумме этих высот, т. е. 3*2=6.

‍ Значит, S‍1 = 3al = 18

‍ПустьS --‍ площадь основания призмы. Площадь ортогональной проекции основания призмы на плоскость, перпендикулярную боковым рёбрам, равна площади перпендикулярного сечения, делённой на косинус угла между плоскостями основания и перпендикулярного сечения. Этот угол равен углу между боковым ребром и высотой призмы, т. е. 60‍∘.

‍ Поэтому

S2= 2√3

Следовательно, площадь полной поверхности призмы равна

 = 18 + 4√3

Answer 2

Problem Analysis

We are given a right triangular prism with a slant height of 2 units, a lateral edge of 3 units, and an inclination angle of 30 degrees with respect to the base plane. We need to find the total surface area of the prism.

Solution

To find the total surface area of the prism, we need to calculate the areas of the two triangular bases and the three rectangular lateral faces, and then sum them up.

# Base Triangles

The base of the prism is a right triangle. We can use the formula for the area of a triangle to find the area of each base triangle. The formula for the area of a triangle is given by:

Area = (1/2) * base * height

In this case, the base of the triangle is the length of the lateral edge, which is 3 units. The height of the triangle can be found using trigonometry. Since the triangle is right-angled, we can use the sine function to find the height. The formula for the height is given by:

Height = slant height * sin(inclination angle)

Substituting the given values, we can calculate the height of each base triangle.

# Lateral Faces

The lateral faces of the prism are rectangles. The length of each lateral face is equal to the length of the lateral edge, which is 3 units. The width of each lateral face can be found using trigonometry. Since the lateral edge is inclined at an angle of 30 degrees with respect to the base plane, we can use the cosine function to find the width. The formula for the width is given by:

Width = lateral edge * cos(inclination angle)

Substituting the given values, we can calculate the width of each lateral face.

# Total Surface Area

To find the total surface area of the prism, we sum up the areas of the two base triangles and the three lateral faces.

Let's calculate the values step by step.

Step 1: Calculate the height of each base triangle using the formula: Height = slant height * sin(inclination angle). - Slant height = 2 units - Inclination angle = 30 degrees

Step 2: Calculate the width of each lateral face using the formula: Width = lateral edge * cos(inclination angle). - Lateral edge = 3 units - Inclination angle = 30 degrees

Step 3: Calculate the area of each base triangle using the formula: Area = (1/2) * base * height. - Base = lateral edge = 3 units - Height (from Step 1)

Step 4: Calculate the area of each lateral face using the formula: Area = length * width. - Length = lateral edge = 3 units - Width (from Step 2)

Step 5: Calculate the total surface area by summing up the areas of the two base triangles and the three lateral faces.

Let's perform the calculations.

Step 1: Calculate the height of each base triangle: - Height = 2 * sin(30°) = 2 * 0.5 = 1 unit

Step 2: Calculate the width of each lateral face: - Width = 3 * cos(30°) = 3 * 0.866 = 2.598 units (rounded to 3 decimal places)

Step 3: Calculate the area of each base triangle: - Area = (1/2) * base * height = (1/2) * 3 * 1 = 1.5 square units

Step 4: Calculate the area of each lateral face: - Area = length * width = 3 * 2.598 = 7.794 square units (rounded to 3 decimal places)

Step 5: Calculate the total surface area: - Total Surface Area = 2 * Area of Base Triangles + 3 * Area of Lateral Faces - Total Surface Area = 2 * 1.5 + 3 * 7.794 = 2.5 + 23.382 = 25.882 square units (rounded to 3 decimal places)

Therefore, the total surface area of the given prism is approximately 25.882 square units.

Note: The calculations have been rounded to 3 decimal places for simplicity.