Question

Основание пирамиды служит квадрат одно из боковых ребер перпендикулярно к плоскости оснований, наибольшее боковое ребро равно 6дм наклонено к основанию под углом 45° найти площадь основания

Answer 1

Диагональ основания равна 6*cos 45 = 6*√2 / 2 = 3√2.
Сторона основания равна 3√2 / √2 = 3 дм.
S = 3*3 = 9 дм².

Answer 2

Finding the Area of the Base of a Pyramid

To find the area of the base of a pyramid, we need to know the dimensions of the base. In this case, we are given that one of the lateral edges of the pyramid is perpendicular to the plane of the base, and the length of the longest lateral edge is 6 dm, inclined at an angle of 45° to the base.

To calculate the area of the base, we need to determine the length of one side of the base. Since the given lateral edge is inclined at a 45° angle to the base, we can use trigonometry to find the length of the base.

Let's denote the length of one side of the base as x. Since the given lateral edge is inclined at a 45° angle, we can use the sine function to relate the length of the lateral edge to the length of the base:

sin(45°) = x / 6

Simplifying the equation, we have:

x = 6 * sin(45°)

Using a calculator, we find that sin(45°) = 0.7071. Substituting this value into the equation, we get:

x = 6 * 0.7071

Calculating the value, we find that x ≈ 4.2426.

Now that we know the length of one side of the base (x ≈ 4.2426 dm), we can calculate the area of the base by squaring this length:

Area of the base = x^2

Substituting the value of x, we have:

Area of the base ≈ (4.2426 dm)^2

Calculating the value, we find that Area of the base ≈ 18 dm^2.

Therefore, the approximate area of the base of the pyramid is 18 square decimeters.

Please note that the calculations provided are approximate due to the rounding of decimal places.