Exercise XV num 12 - Calculus Made Easy
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A bucket of given capacity has the shape of a horizontal isosceles triangular prism with the apex underneath, and the opposite face open. Find its dimensions in order that the least amount of iron sheet may be used in its construction. My approach: Lets say all dimensions are each $ > 0$ ; Volume of the bucket - $V=frac{lwh}{2}$ ,where l-length of toblerone, w-width, h-height; Total exterior area of the bucket - $A= 2frac {hw}{2}+2l(frac{w^2}{4}+h^2)^frac{1}{2} Rightarrow$ $A= hw+frac{4V}{hw}(frac{w^2}{4}+h^2)^frac{1}{2} $ ; $frac{partial A}{partial h} = w - frac {4V}{h^2w}(frac{w^2}{4}+h^2)^frac{1}{2}+frac {4V}{w(frac{w^2}{4}+h^2)^frac{1}{2}}$ ; $frac{partial A}{partial w} = h - frac {4V}{hw^2}(frac{w^2}{4}+h^2)^frac{1}{2}+frac {V}{h(frac{w^2}{4}+h^2)^frac{1}{2}}$ ; Is this the right path to conti...
