Two spheres of radius 6 cm and 4cm are inscribed in a right circular cone.The bigger sphere touches the smaller one and also the base of cone .find the height of the cone?
Draw a cross-section diagram of the situation.
the sides of the cone would be tangent to both circles.
draw radii of both circles to a side of the cone, with the right angle as they meet the side
Let the remaining distance from the top of the smaller circle to the vertex of the cone be x cm
I see two similar right-angled triangles, so we can use ratios
(x+4)/4 = (x+14)/6
6x + 24 = 4x + 56
2x = 32
x = 16
So the height = 12 + 8 + x = 36 cm
To find the height of the cone, we can use the concept of similar triangles.
Let's consider the right circular cone and draw a line from the vertex to the center of the base. This line is the height of the cone.
Now, let's focus on the cross-section of the cone. We have two spheres inscribed in the cone. The bigger sphere touches both the smaller sphere and the base of the cone. This implies that the line drawn from the vertex to the center of the bigger sphere passes through the center of the smaller sphere.
We can create a right triangle within the cross-section by drawing a line from the center of the smaller sphere to the center of the cone's base, as well as a line perpendicular to the base connecting the centers of both spheres. The length of this line is the sum of the radii of both spheres.
In this right triangle, the radius of the bigger sphere (6 cm) is the hypotenuse, the radius of the smaller sphere (4 cm) is one of the legs, and the length of the line from the vertex to the center of the bigger sphere is the other leg.
By using the Pythagorean theorem, we can solve for the length of the line from the vertex to the center of the bigger sphere, which is the height of the cone.
Let's solve it step by step:
1. Let h be the height of the cone.
2. According to the Pythagorean theorem, we have (6 cm)^2 = (h + 4 cm)^2 + (4 cm)^2.
3. Simplifying the equation, we get 36 cm^2 = h^2 + 8h + 16 cm^2 + 16 cm^2.
4. Combining like terms, we have 36 cm^2 = h^2 + 8h + 32 cm^2.
5. Rearranging the equation, we get h^2 + 8h - 4 cm^2 = 0.
6. To solve this quadratic equation, we can use the quadratic formula: h = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = 8, and c = -4.
7. Plugging in the values, we get h = (-8 ± √(8^2 - 4(1)(-4))) / (2(1)).
8. Simplifying further, we get h = (-8 ± √(64 + 16)) / 2.
9. h = (-8 ± √80) / 2.
10. h = (-8 ± 8.944) / 2.
11. h = (-8 + 8.944) / 2 or h = (-8 - 8.944) / 2.
12. h = 0.944 / 2 or h = -16.944 / 2.
13. h = 0.472 or h = -8.472.
Since a height cannot be negative, the height of the cone is approximately 0.472 cm.
Therefore, the height of the cone is approximately 0.472 cm.