Final answer:
The height of the cylinder is 8mm.
Step-by-step explanation:
To find the height of the cylinder, we need to first find the surface area of the sphere. The surface area of a sphere is given by the formula: 4πr^2. Since the diameter of the sphere is 16mm, the radius is half of that, which is 8mm. Plugging the value of the radius into the formula, we get: 4π(8^2) = 256π mm^2.
Next, we need to find the surface area of the cylinder. The base diameter of the cylinder is equal to the diameter of the sphere, which is 16mm. Therefore, the radius of the cylinder is also 8mm. The surface area of the cylinder is given by the formula: 2πrh + 2πr^2. Since the height is unknown, we'll use a variable 'h'.
Since the surface area of the sphere is equal to the surface area of the cylinder, we can set up an equation: 256π = 2πrh + 2πr^2. Plugging in the values, we get: 256π = 2π(8)(h) + 2π(8^2).
Cancelling out the common factor of 2π, we have the equation: 128 = 8h + 64. Subtracting 64 from both sides of the equation, we get: 64 = 8h. Dividing both sides by 8, we find that the height of the cylinder is: h = 8mm.