Help solve pleaseAn international consortium plans to build the fanciest steakhouse of the world, Westmount Cove,

at the foothill of Montreal's Westmount. The restaurant will look like a cave: part of the

Westmount will be excavated, and the three quarter of the restaurant will be build inside the

excavation. A London$-$based architect does the design. For his initial draft design, he imagined the

exterior structure as a paraboloid with the equation $z=\frac{-(x^2+y^2)}{25},$ where the centre of the origin is

located at the top of the structure. The surface of this structure inside the excavation is built by

concrete with the constant thickness of $20cm.$ The other quarter of the exterior is outside the

excavation and covered by glass with the same $20cm$ thickness as the conerete wall. To maintain

structural stability, the structure must include a central loading column. The architect designs this

column as a hollow concrete cone so that the space inside the cone can be used as a bar on the

ground floor, and private spaces, and kitchen$/$storage on higher levels $-$ see the figure below,

$($a$)$

$($b$)$

Figure. $($a$)$ The $3$D shape and $($b$)$ the plan of Wesmount Cove, an envisioned steakhouse in

Westmount, Montreal, competing to mark the fanciest and finest steakhouse in the world. The red

column and the black exterior are built by concrete. The opening exterior is covered by glass.

To make the structure safe from design standpoint, the structural engineer ${?}^1$ from Montreal

determines that the radius of the hollow cone column must be at least one eighth of the radius for

the base for paraboloid and its wall thickness should be at least $35cm.$ The crowns of the cone and

the paraboloid meet the same point, which is the top of the structure. Assuming again this point as

the centre of the origin, the architect used $z=-25\mathrm{.}6\sqrt[\phantom{2}]{x^2+y^2}$ as the equation for the column.

While the consortium's board likes the design, they have a major hesitation regarding the cost of

constructing such a magnificent design. They agreed in principle with the design but want to cap

the structural cost for constructing the concrete and glass walls and the loading column to $$2M$ at

the very most. This cost should include the price of the concrete for the exterior walls inside the

excavation as well as the hollow cone column, along with the price of the glass wall. The price of

concrete is $$\frac{500}{m^3}$ and includes the construction labor and transportation costs too. The unit price

for exterior glass wall can be calculated by $$500$ times the unit area multiplied by its thickness,

multiplied by its curvature plus a fixed $$500$ for each meter of height. The only supplier of a such

a glass wall is an Italian manufacturer based in Milan that uses the formula of $=$

to calculate the curvature of its glass production. The quoted price of glass wall

$(1+(\frac{df}{dx}{)}^2+(\frac{df}{dy}{)}^2{)}^{\frac{1}{2}}$

also includes labor work, transportation, and installation cost.

Using the theory of multiple integral and your general knowledge of calculus as a whole:

a$)$ Find the price of all concrete works including three quarter of outer paraboloid and the

whole inner cone as a function of radius of paraboloid $R^{**}$;$R^{**}=8R(30\%{)}^2.$

b$)$ Find the price of glass wall as a function of $R^{**}(30\%).$

c$)$ What would be the critical value of $R^{**}$ after which the price exceeds $$2M.$ Is this critical

design feasible? Look at the feasibility of design from the perspective of the guest space

$($i$.$e$.,$ three quarter of the base of paraboloid minus the base area of the cone$)$ and the height

of the celling. Do you think that the architect can convince the board that his design will

be certainly cost less than $$2M(30\%{)}^3?$

d$)$ Before taking the design to the board, the architect needs to finalize the shape of the cone

and paraboloid. Imagine the general equation of cone as $z=-a\sqrt[\phantom{2}]{x^2+y^2}$ and for the

paraboloid as $z=\frac{-(x^2+y^2)}{b}.$ Given the fact the radius of cone is one eighth of the radius of

the paraboloid, find $b$ as a function of $R^{**}$ and $a,$ knowing that the top of the cone and

paraboloid should meet at their very top $(10\%).$