Question

. В основании прямой призмы лежит равнобедренный треугольник с основанием 6 см и углом при вершине 120* Диагональ боковой грани содержащей основание равнобедренного треугольника 10 см найдите площадь боковой поверхности .

Answer 1

S=P*h; P=2a+c; c=10sm; a=c*ctg60=10*$\frac {\sqrt{3}}{3}$; h=K(10^2-6^2)=K(36)=6; P=2*10*$\frac {\sqrt{3}}{3}$+10=10(2*$\frac {\sqrt{3}}{3}$+1).
S=10(2*$\frac {\sqrt{3}}{3}$+1)*6=60(2*$\frac {\sqrt{3}}{3}$+1)

K(10^2-6^2) - это корень квадратный из выражения в скобках

Answer 2

Problem Analysis

We are given a right prism with a base that is an isosceles triangle with a base of 6 cm and a vertex angle of 120 degrees. The diagonal of one of the lateral faces containing the base is 10 cm. We need to find the lateral surface area of the prism.

Solution

To find the lateral surface area of the prism, we need to find the perimeter of the base and multiply it by the height of the prism.

Let's start by finding the perimeter of the base. Since the base is an isosceles triangle, we can use the formula for the perimeter of an isosceles triangle, which is given by:

Perimeter = 2 * base + side

In this case, the base of the triangle is 6 cm. To find the length of the side, we can use the sine rule for triangles, which states that:

side / sin(vertex angle) = base / sin(base angle)

In our case, the vertex angle is 120 degrees and the base angle is (180 - 120) / 2 = 30 degrees. Plugging in the values, we can solve for the length of the side:

side / sin(120) = 6 / sin(30)

Simplifying the equation, we get:

side = (6 * sin(120)) / sin(30)

Now that we have the length of the side, we can calculate the perimeter of the base:

Perimeter = 2 * 6 + side

Next, we need to find the height of the prism. The height is the length of the diagonal of the lateral face containing the base. In this case, the length of the diagonal is given as 10 cm.

Finally, we can calculate the lateral surface area of the prism by multiplying the perimeter of the base by the height:

Lateral Surface Area = Perimeter * Height

Let's calculate the values step by step:

1. Calculate the length of the side: - side = (6 * sin(120)) / sin(30) - side ≈ 10.3923 cm

2. Calculate the perimeter of the base: - Perimeter = 2 * 6 + side - Perimeter ≈ 22.7846 cm

3. Calculate the lateral surface area: - Lateral Surface Area = Perimeter * Height - Lateral Surface Area ≈ 22.7846 cm * 10 cm - Lateral Surface Area ≈ 227.846 cm²

Therefore, the lateral surface area of the prism is approximately 227.846 cm².

Answer

The lateral surface area of the prism is approximately 227.846 cm².