To celebrate the acquisition of Styria in 1261, Ottokar ll sent hunters into the

Question

To celebrate the acquisition of Styria in 1261, Ottokar ll sent hunters into the Bohemian woods to capture a unicorn. To display the unicorn at court, the king built a rectangular cage. The material for 3 sides of the cage cost 3 known ducats/running cubit, while the 4^th side was to be gilded and cost 51 ducats/running cubil. In 1261 it as well the that a happy uicorn requires an area of 2025 square cubits in which to live. Find the dimensions that would keep unicorn happy at the lowest cost. What is the lowest cost to build the cage given these requirements?

Explanation / Answer

Since we are given a rectangular cage with are area of 2025 ==> x*y = 2025 (Area = Length*Breadth)

Now the Cost Equation Can be wrtten as C(x) = 3x+3x+3y+51y

Now we know y = 2025/x

===> C(x) = 6x + 3(2025/x) + 51(2025/x) = 6x + 103275/x

Now to find the lowest cost to build the cage is given by C'(x) = 0

==> C'(x) = 6 - 103275/x2 and ===> 6 - 103275/x2 = 0 ==> X = 131. 19641.

Now finding the double derivative of above Cost equation

==> C''(x) = 2*103275/x3 = 206550/x3

Now C"(131.19641) = 206550/(131.19641)3 = 0.09146592804

Since the second derivative is positive, the cost function is concave up at x = 131.19641 so it is a minimum. We now calculate the minimum cost as C(131.19641) = 6(131.19641) + 103275/(131.19641) = 1574.35701161 ducats.

Now Similarly y = 2025/x = 2025/131.19641 = 15.4348735609

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