Final answer:
To find the height of the cone, we can compare the volumes of the cone and the hemisphere. The height of the cone is found to be 31.5 cm. The surface area of the toy is approximately 9035.77 cm².
Step-by-step explanation:
To find the height of the cone, we need to compare the volume of the cone to the volume of the hemisphere. Let's assume the height of the cone is 'h'.
The volume of the cone is given by V = 1/3 * π * r^2 * h, where r is the radius of the base of the cone.
Given that the volume of the cone is 2/3 of the volume of the hemisphere, we have 1/3 * π * 21^2 * h = 2/3 * (2/3 * π * (21/2)^3).
Simplifying, we get h = 6 * (21/2)^3 / 21^2 = 3 * 21 / 2 = 31.5 cm.
To find the surface area of the toy, we need to calculate the surface area of the hemisphere and the surface area of the cone, and then add them together.
The surface area of the hemisphere is given by A = 2 * π * r^2, where r is the radius of the hemisphere.
The surface area of the cone is given by A = π * r * l, where r is the radius of the base of the cone and l is the slant height.
To find the slant height, we can use the Pythagorean theorem, l = sqrt(h^2 + r^2).
Substituting the values, we have A = 2 * π * (21/2)^2 + π * 21 * sqrt((31.5)^2 + (21/2)^2) = 2209 π + 661.5 π = 2870.5 π.
Using π ≈ 3.142, we get A ≈ 9035.77 cm².