The volume of a cone is 25/3 π cm3. What is the volume of a sphere if its radius is the same as the cone’s and the height of the cone is equal to the sphere’s diameter?(1 point)
Responses
25/6 π cm3
Start Fraction 25 over 6 End Fraction pi cm cubed
25/2 π cm3
Start Fraction 25 over 2 End Fraction pi cm cubed
The correct answer is 25/6 π cm3 or (Start Fraction 25 over 6 End Fraction pi cm cubed). This is because the volume of a cone is calculated using the formula V = (1/3)πr^2h, where r is the radius and h is the height. Since the height of the cone is equal to the sphere's diameter, we can say that the height of the cone (h) is equal to 2r. Substituting this into the formula, we get V = (1/3)πr^2(2r) = (2/3)πr^3. We are given that V = 25/3 π cm3, so we can solve for r^3 by multiplying both sides of the equation by 3/2: r^3 = (25/3 π cm3) * (3/2) = 25/2 π cm3. Finally, taking the cube root of both sides, we find that r = (25/2 π cm3)^(1/3). The volume of a sphere is given by the formula V = (4/3)πr^3, so substituting the value of r, we get V = (4/3)π((25/2 π cm3)^(1/3))^3 = (4/3)π(25/2 π cm3) = 25/6 π cm3.
