Question

Длина диогонали правильной четырехугольной призмы равна 9см. площадь ее полной поверхности 144 см^2. Натиск длину стороны основания и длину бокового ребра

Answer 1

Обозначим сторону основания а, а высоту призмы H. Тогда Sп.п = 4aH+ 2a^2  H= √81 - 2a^2   Имеем 144 = 4a√81 - 2a^2 + 2a^2  Или, возведя в квадрат и перенеся в одну сторону, получим: 36a^4 - 1872a^2 + 20736 = 0 Разделив на 36 и обозначив a^2 = t имеем t^2 - 52t + 576 =0 Корни этого уравнения 16 и 36. Следовательно имеем два решения. Первая призма имеет сторону основания 4 и высоту 7. Второй вариант, основание 6 см, высота 3см  Оба варианта удовлетворяют условию.

Answer 2

Calculation of the Length of the Diagonal of a Prism

To find the length of the diagonal of a rectangular prism, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the case of a rectangular prism, the diagonal is the hypotenuse of a right triangle formed by the length, width, and height of the prism.

Let's denote the length of the diagonal as d, the length of the prism as l, the width as w, and the height as h.

According to the given information, the length of the diagonal is 9 cm. We can set up the following equation using the Pythagorean theorem:

l^2 + w^2 + h^2 = d^2

Substituting the given values, we have:

l^2 + w^2 + h^2 = 9^2

Simplifying further, we have:

l^2 + w^2 + h^2 = 81

Calculation of the Surface Area of the Prism

The surface area of a rectangular prism can be calculated by finding the sum of the areas of its six faces. Each face of the prism is a rectangle, and the area of a rectangle is given by the product of its length and width.

Let's denote the surface area of the prism as A, the length of the prism as l, the width as w, and the height as h.

According to the given information, the surface area of the prism is 144 cm^2. We can set up the following equation:

2lw + 2lh + 2wh = A

Substituting the given values, we have:

2lw + 2lh + 2wh = 144

Simplifying further, we have:

lw + lh + wh = 72

Calculation of the Length of the Base Side and the Length of the Lateral Edge

To find the length of the base side and the length of the lateral edge, we need additional information or equations. Unfortunately, the given information does not provide enough details to determine these values.

If you have any additional information or equations related to the length of the base side and the length of the lateral edge, please provide them, and I will be happy to help you further.