What is the maximum possible volume of a cone that has a slant height of 10 cm?
The volume of a cone with perpendicular height h and radius of the base r is given by `V = (1/3)*pi*r^2*h`
If a cone has a slant height (S), perpendicular height (h) and radius (r), the three are related by `S^2 = h^2 + r^2`
Here S = 10
If the height is x, the radius is: `sqrt(100 - x^2)`
The volume of the cone is `V = (1/3)*pi*(100 - x^2)*x`
To maximize V, solve `(dV)/(dx) = 0`
=> `(1/3)*pi*100 - (1/3)*pi*3*x^2 = 0`
=> `x^2 = 100/3`
=> `x = 10/sqrt 3`
The height of the cone is `10/sqrt 3` and the radius is `sqrt(200/3)` . The volume of the cone is: `(1/3)*pi*(200/3)(10/sqrt 3) = 403.06`
The maximum possible volume of a cone with slant height 10 cm is 403.06 cm^3.