Chef Potato needs to be converted from a two-dimensional shape to a three-dimensional
object and then enlarged to 300% of his original size for an advertising campaign. The threedimensional Chef Potato is similar in shape to a capsule made of a half-sphere at the top and a
half-sphere at the bottom and a cylinder in between.
a. Determine the surface area of the enlarged three-dimensional Chef Potato.
b. Determine the volume of the enlarged three-dimensional Chef Potato.
thank you very much for any responses.
so basically the entire figure consists of a cylinder and a sphere.
For volume, that's no problem.
For the area, ignore the ends of the cylinder, which are hidden.
That is, if the cylinder (and sphere) have radius r, and the cylinder has height h, then the volume is
v = 4/3 πr^3 + πr^2h = πr^2 (4/3 r + h)
a = 4πr^2 + 2πrh = 2πr(2r+h)
You have given no actual dimensions, but recall that if the linear size is increased by a factor of 3,
the area grows by a factor of 3^2
the volume grows by a factor of 3^3
thank you very much Steve
thank you Steve
To determine the surface area and volume of the enlarged three-dimensional Chef Potato, we need to understand the shape and dimensions of each component (half-spheres and cylinder) in the capsule.
Let's start with the surface area.
a. Surface Area:
The Chef Potato consists of a half-sphere at the top, a half-sphere at the bottom, and a cylinder in between. To find the surface area, we'll calculate the surface area of each component separately and then add them up.
1. Half-Sphere at the Top:
The surface area of a half-sphere is given by the formula: A = 2πr^2, where r is the radius.
Since the Chef Potato is enlarged by 300%, the radius will be increased by the same factor. Let's assume the original radius of the half-sphere is r.
After enlarging by 300%, the new radius (r') of the half-sphere will be r' = r × 3.
The surface area of the enlarged half-sphere at the top is A_top = 2π(r')^2.
2. Half-Sphere at the Bottom:
Similar to the top half-sphere, the surface area can be calculated using the same formula: A_bottom = 2π(r')^2.
3. Cylinder in Between:
The surface area of a cylinder is given by the formula: A = 2πrh, where r is the radius and h is the height.
Since the Chef Potato is enlarged by 300%, both the radius and height of the cylinder will be increased by the same factor.
Let's assume the original radius and height of the cylinder are r and h, respectively.
After enlarging by 300%, the new radius (r') and new height (h') will be r' = r × 3 and h' = h × 3.
The surface area of the enlarged cylinder is A_cylinder = 2π(r')h'.
Finally, to find the total surface area of the Chef Potato, we add the surface areas of the top half-sphere, bottom half-sphere, and the cylinder:
Total Surface Area = A_top + A_bottom + A_cylinder
b. Volume:
To calculate the volume, we'll consider each component separately and then add them up.
1. Half-Sphere at the Top:
The volume of a half-sphere is given by the formula: V = (4/3)πr^3, where r is the radius.
The volume of the enlarged half-sphere at the top is V_top = (4/3)π(r')^3.
2. Half-Sphere at the Bottom:
The volume of the enlarged half-sphere at the bottom is V_bottom = (4/3)π(r')^3.
3. Cylinder in Between:
The volume of a cylinder is given by the formula: V = πr^2h, where r is the radius and h is the height.
The volume of the enlarged cylinder is V_cylinder = π(r')^2h'.
Finally, to find the total volume of the Chef Potato, we add the volumes of the top half-sphere, bottom half-sphere, and the cylinder:
Total Volume = V_top + V_bottom + V_cylinder.
By substituting the appropriately enlarged dimensions and performing the calculations, you can determine the surface area and volume of the enlarged three-dimensional Chef Potato.