Question

основанием четырехугольной пирамиды является прямоугольник с диагональю 4 см и углом 60 градусовмежду диагоналями.Боковые ребра наклонены к плоскости основания под углом 45 градусов.Наити объём пирамиды

Answer 1

Finding the Volume of a Pyramid

To find the volume of a pyramid, we need to know the area of the base and the height of the pyramid. In this case, we are given that the base of the pyramid is a rectangle with a diagonal of 4 cm and an angle of 60 degrees between the diagonals. The lateral edges of the pyramid are inclined to the base at an angle of 45 degrees.

Let's break down the problem step by step:

1. Finding the Length and Width of the Rectangle Base: - Since the base of the pyramid is a rectangle, we can use the given diagonal and the angle between the diagonals to find the length and width of the rectangle. - The diagonal of the rectangle is 4 cm, and the angle between the diagonals is 60 degrees. - Using trigonometry, we can find the length and width of the rectangle. - Let's denote the length of the rectangle as L and the width as W. - We can use the formula: L = 2Wsin(60°), where sin(60°) = √3/2. - Simplifying the equation, we get: L = √3W. - We also know that the diagonal of the rectangle is 4 cm, so we can use the Pythagorean theorem to find the relationship between L and W. - The Pythagorean theorem states that L^2 + W^2 = 4^2. - Substituting the value of L from the previous equation, we get: (√3W)^2 + W^2 = 4^2. - Simplifying the equation, we get: 3W^2 + W^2 = 16. - Combining like terms, we get: 4W^2 = 16. - Dividing both sides by 4, we get: W^2 = 4. - Taking the square root of both sides, we get: W = 2. - Therefore, the width of the rectangle base is 2 cm. - Substituting this value back into the equation L = √3W, we get: L = √3(2) = 2√3. - Therefore, the length of the rectangle base is 2√3 cm.

2. Finding the Height of the Pyramid: - To find the height of the pyramid, we need to consider the lateral edges that are inclined to the base at an angle of 45 degrees. - The height of the pyramid is the perpendicular distance from the apex (top) of the pyramid to the base. - Since the lateral edges are inclined at an angle of 45 degrees, the height forms a right triangle with the inclined edge and the base. - The inclined edge is the hypotenuse of the right triangle, and the height is the opposite side. - Let's denote the height of the pyramid as H. - We can use trigonometry to find the relationship between the height and the inclined edge. - The inclined edge is the hypotenuse, so we can use the formula: H = Lsin(45°), where sin(45°) = 1/√2. - Substituting the value of L from earlier, we get: H = 2√3 * (1/√2) = √3. - Therefore, the height of the pyramid is √3 cm.

3. Finding the Volume of the Pyramid: - Now that we have the length, width, and height of the pyramid, we can find its volume. - The volume of a pyramid is given by the formula: Volume = (1/3) * Base Area * Height. - The base area of the pyramid is the area of the rectangle, which is Length * Width. - Substituting the values, we get: Base Area = (2√3 cm) * (2 cm) = 4√3 cm^2. - Substituting the values into the volume formula, we get: Volume = (1/3) * (4√3 cm^2) * (√3 cm). - Simplifying the equation, we get: Volume = (4/3) * (√3 cm)^2 * (√3 cm). - Simplifying further, we get: Volume = (4/3) * 3 cm^3. - Finally, the volume of the pyramid is: Volume = 4 cm^3.

Therefore, the volume of the given pyramid is 4 cm^3.