Question

Background: A company manufactures solid objects known as gizmoids made of variable density plastic.

In the company's design specifications, a gizmoid is modelled as a solid region V satisfying the conditions:
x ²+y²+z² ≤ 1 and √3(x² + y²).
The mass per unit volume (mass density) of V is
rho(x,y,z)=10z
The point of the gizmoid at (0,0,0) is referred to as the tip.
The company also manufactures the much sought-after deluxe gizmoids. These are identical to regular gizmoids, except that the part of the gizmoid more than 0.9 units from the tip is made
of gold rather than plastic. [Note that the set of all points in a gizmoid that are exactly 0.9 units from the tip form part of the surface of a sphere of radius 0.9 . This means that the part of a
deluxe gizmoid made of gold has two spherical boundary surfaces.]
(a) Sketch V.
(b) Calculate the mass of a gizmoid, using cylindrical coordinates.
(c) Calculate the mass of a gizmoid, using spherical coordinates.
(d) Determine the exact volume of gold required to manufacture 1,000 deluxe gizmoids.
(e) Assuming that gold has a constant density (mass per unit volume) of 19, use the MATLAB symbolic toolbox to calculate the mass of a deluxe gizmoid. [Hint: Use spherical coordinates.]

Answer 1

The density of gold, 19.3 g/cm³, is used to calculate the mass based on the volume of gold in the deluxe gizmoids. The volume of the gold part is calculated using geometrical methods, then multiplied by the density to find the required mass for 1,000 units.

Step-by-step explanation:

The density of gold is crucial when calculating the mass of gold objects based on their volume. For gold, this density is typically 19.3 g/cm³. This means for any gold object, if we know the volume in cubic centimeters, we can find its mass in grams by multiplying the volume by 19.3. In the context of manufacturing deluxe gizmoids where part of the structure is replaced with gold, a precise calculation of the gold volume is essential to determine the total mass of the deluxe gizmoids.

To accomplish the task of determining the required mass of gold for manufacturing 1,000 deluxe gizmoids, one must first establish the volume of the gold portion using relevant geometrical formulas. In the case mentioned, where the gold comprises two spherical caps of a gizmoid, the volume can be found using spherical coordinates for the integration. After finding the volume of gold, the mass can be calculated by multiplying it by the density of gold.