the radius of a ball which fits into a cone with a radius of 2cm & a height of 5cm
To determine the radius of a ball that fits into a cone, we need to compare the volumes of the two objects. The volume of a cone can be calculated using the formula V_cone = (1/3) * π * r^2 * h, where r is the radius of the cone and h is the height of the cone. The volume of a sphere (or ball) is given by V_sphere = (4/3) * π * r^3, where r is the radius of the sphere.
In this case, the given values for the cone are r_cone = 2 cm and h_cone = 5 cm. Let's denote the radius of the sphere as r_sphere.
Since the ball must fit inside the cone, the volume of the sphere should be smaller than or equal to the volume of the cone. Therefore, we can set up an equation and solve for r_sphere:
V_sphere ≤ V_cone
Substituting the formulas for the volumes:
(4/3) * π * r_sphere^3 ≤ (1/3) * π * r_cone^2 * h_cone
Canceling out π and multiplying both sides by 3:
4 * r_sphere^3 ≤ r_cone^2 * h_cone
Now, substitute the given values:
4 * r_sphere^3 ≤ 2^2 * 5
4 * r_sphere^3 ≤ 20
Dividing both sides of the inequality by 4:
r_sphere^3 ≤ 5
To isolate r_sphere, take the cube root of both sides:
r_sphere ≤ ∛5
Thus, the radius of the ball that fits into the given cone is less than or equal to the cube root of 5, or approximately 1.71 cm.