Question

Висота конуса дорівнює 15 см, а радіус його основи - 10 см. На відстані 3 см від вершини конуса проведено площину, перпендикулярну до осі, яка перетинає бічну поверхню по колу. Знайдіть радіус перерізу та довжину кола, що утворилося в перерізі

Answer 1

Радіус на відстані 3 см від вершини:

10/15*3=2/3*3=2см

Довіжина кола 2*pi*R:

2*2*pi=4*pi≈12,57 см

Answer 2

Problem Analysis

We are given the height of a cone (15 cm) and the radius of its base (10 cm). A plane is drawn 3 cm away from the vertex of the cone, perpendicular to its axis, and it intersects the lateral surface of the cone in a circle. We need to find the radius of this circle and the length of the circumference.

Solution

To solve this problem, we can use the properties of similar triangles. Let's consider the following diagram:

                A
               /|
              / |
             /  |
            /   | h = 15 cm
           /    |
          /     |
         /      |
        /       |
       /        |
      /         |
     /          |
    /           |
   /            |
  /             |
 /              |
B---------------C

In the diagram above, the cone is represented by the triangle ABC. The vertex of the cone is A, the base of the cone is the circle with center C and radius r, and the point where the plane intersects the lateral surface of the cone is B.

We can see that triangle ABC is similar to triangle ADB, where D is the point where the plane intersects the base of the cone. This is because both triangles share the angle at A and have a right angle at D.

Using the properties of similar triangles, we can set up the following proportion:

AB / AD = AC / AB

Since AB = 3 cm and AD = 15 cm, we can substitute these values into the proportion:

3 / 15 = AC / 3

Simplifying the equation, we find:

AC = 9 cm

Therefore, the radius of the circle formed by the intersection of the plane and the lateral surface of the cone is 9 cm.

To find the length of the circumference of this circle, we can use the formula for the circumference of a circle:

C = 2πr

Substituting the value of the radius (9 cm) into the formula, we find:

C = 2π * 9 = 18π cm

Therefore, the length of the circumference of the circle formed by the intersection of the plane and the lateral surface of the cone is 18π cm.

Summary

In summary, the radius of the circle formed by the intersection of the plane and the lateral surface of the cone is 9 cm, and the length of the circumference of this circle is 18π cm.