Question 321700
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Let *[tex \Large w] represent the width of the prism.  Let *[tex \Large l] represent the length of the prism.


Given a height of 2, the surface area of the prism as a function of length and width is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ SA(w,l)\ =\ 2wl\ +\ 2\,\cdot\,2l\ +\ 2\,\cdot\,2w]


Solve for *[tex \Large l] in terms of *[tex \Large w] and the given surface area of 36.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2wl\ +\ 2\,\cdot\,2l\ +\ 2\,\cdot\,2w\ =\ 36]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ wl\ +\ 2l\ +\ 2w\ =\ 18]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ l(w\ +\ 2)\ =\ 18\ -\ 2w]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ l\ =\ \frac{18\ -\ 2w}{w\ +\ 2}]


The volume of the prism is the length times the width times the height:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ V(w)\ =\ \left(\frac{18\ -\ 2w}{w\ +\ 2}\right)\,\cdot\,w\,\cdot\,2\ =\ \frac{36w\ -\ 4w^2}{w\ +\ 2}]






John
*[tex \LARGE e^{i\pi} + 1 = 0]
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