Question

В шар с радиусом 5 вписана правильная треугольная пирамида с высотой 8 Найдите радиус сечения шара плоскостью проходящей через основание пирамиды

Answer 1

Смотри во вложении.

Answer 2

Finding the Radius of the Sphere

To find the radius of the section of the sphere passing through the base of the pyramid, we need to consider the properties of the pyramid and the sphere.

Given: - A sphere with a radius of 5 is inscribed in a regular triangular pyramid. - The height of the pyramid is 8.

To find the radius of the section, we can use the following steps:

1. Determine the height of the triangular pyramid. - The height of the pyramid is given as 8.

2. Find the length of the base of the triangular pyramid. - Since the pyramid is regular, the base is an equilateral triangle. - The height of an equilateral triangle divides the base into two equal segments, forming two right-angled triangles. - The length of each segment is half the length of the base. - To find the length of the base, we can use the Pythagorean theorem.

3. Use the Pythagorean theorem to find the length of the base. - Let's assume the length of the base is b. - The height of each right-angled triangle is b/2. - The hypotenuse of each right-angled triangle is the radius of the sphere, which is 5. - Applying the Pythagorean theorem, we have: (b/2)^2 + 5^2 = b^2.

4. Solve the equation to find the length of the base. - Expanding the equation, we have: b^2/4 + 25 = b^2. - Multiplying both sides by 4, we get: b^2 + 100 = 4b^2. - Rearranging the equation, we have: 3b^2 = 100. - Dividing both sides by 3, we get: b^2 = 100/3. - Taking the square root of both sides, we have: b = sqrt(100/3).

5. Find the radius of the section of the sphere passing through the base. - The radius of the section is equal to the height of the pyramid, which is 8.

Therefore, the radius of the section of the sphere passing through the base of the pyramid is 8.

Please note that the calculations provided above are based on the given information and assumptions.