During her holidays to Victoria, Uschi visited the Heide Museum of Modern Art to

Question

During her holidays to Victoria, Uschi visited the Heide Museum of Modern Art to discover a striking gazebo, situated in the Gardens and Sculpture Park, with a seemingly curved roof. Upon her return to Brisbane, her architect advised that the roof is (mostly) fabricated from straight pieces of timber joists. They sat down together to design a similar gazebo for her arboretu FiGURE Uschi's photograph of the gazebo at the Heide. The gazebo will be located on a site covering 12 metres by 12 metres. The roof will be constructed to fit the surface given by the function 2 200 on the domain [-6.6] × [-6.6], with x, y, and z all measured in metres. Nine timber joists will be cut, and positioned such that they lie on cross sections of the surface. Shadecloth will be laid on top of the joists to produce a curved roof effect Your task is to work through the following steps to determine the length of each joist

Explanation / Answer

Given - Equation of surface = 5/2 + (1/200)(9x2 -4y2)

Equation of vertical plan(e =ax+by=c

line of intersection of cross section and vertical plane=

z(x)=5/2 +1/200(9x2-4((c-ax)/b)2)

=5/2 +1/200((9x2b2-4c2-4a2x2+8acx)/b2)

For joists to be straight

Coeffecients of x2 must be zero

9b2-4c2=0  

Case 1 3b=2a Case 2 3b=-2a

Case 1    3b=2a

z=5/2 + 1/200((8acx-4c2)/b2)

Put 2a=3b

=5/2 + 1/200(12(c/b)x-4(c/b)2)

=5/2 + 1/50(3(c/b)x-4(c/b)2)

put t=c/b

General Equation of Crosssection Z(t) = 5/2 + 1/50(3tx-4t2)

Case 2 3b=-2a

z=5/2 + 1/200((8acx-4c2)/b2)

Using 2a=-3b

=5/2 + 1/200(-12(c/b)x-4(c/b)2)

=5/2 + 1/50(-3(c/b)x-4(c/b)2)

put t=c/b

General Equation of Crosssection Z(t) = 5/2 - 1/50(3tx+4t2)

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