A spherical foam ball, 10 inches in diameter, is used to make a tabletop decoration for a party. To make the decoration sit flat on the table, a horizontal slice is removed from the bottom of the ball, as shown below. If the radius of the flat surface formed by the cut is 4 inches, what is the height of the decoration?
3 , 4 , 5 right triangle with 3 down from center, 5 hypotenuse radius ball center to surface intersection with table, 4 radius of cut
3 + 5 = 8
To find the height of the decoration, we need to understand the relationship between the radius of the flat surface and the height of the spherical foam ball.
Let's assume that the center of the spherical foam ball is at point O. The radius of the ball is given as 10 inches, which means the distance from the center O to any point on the surface of the ball is 10 inches.
When a horizontal slice is made at the bottom of the ball, it forms a circle with a flat surface. The radius of this flat surface is given as 4 inches. Let's assume this flat surface is at a height h from the center O.
We can form a right triangle with the radius of the ball (10 inches), the radius of the flat surface (4 inches), and the height of the decoration (h).
Using the Pythagorean theorem, we have:
(10 inches)^2 = (4 inches)^2 + h^2
Simplifying the equation:
100 inches^2 = 16 inches^2 + h^2
84 inches^2 = h^2
Taking the square root of both sides:
√(84 inches^2) = √h^2
8.17 inches = h
Therefore, the height of the decoration is approximately 8.17 inches.