(Solved): A simple way to model the construction of an oil tanker is to start with a large rectangular sheet ...

A simple way to model the construction of an oil tanker is to start with a large rectangular sheet of steel that is \( x \) feet wide and \( 3 x \) feet long. Now cut a smaller square that ist feet on a side out of each corner of the larger sheet and fold up and weld the sides of the steel sheet to make the structure with no top. The orange rectangle on following diagram represents the sheet of the steel, with dimensions \( 3 x \). Place four purple squares on the orange rectangle that represent the smaller squares (with side length \( t \) ) to be cut out of the steel sheet. Once the sides have been folded up and welded in place, the structure is now resembles a tray with no top. This tray has a height of \( t \), with the dimensions of the base \( (x-2 t) \) by \( (3 x-2 t)= \) which of the following formulas describes the volume of the resulting tray-like structure. \[ \begin{array}{l} V=t(x-t)(3 x-t)=3 t x^{2}-4 t^{2} x+t^{3} \\ V=t(x+2 t)(3 x+t)=3 t x^{2}+7 t^{2} x+2 t^{3} \\ V=t(x-2 t)(3 x-2 t)=3 t x^{2}-8 t^{2} x+4 t^{3} \end{array} \] For a given value of \( x \), the value of \( t \) that maximizes the volume is \( t^{*}= \) True or False: There is a value of \( x \) that maximizes the volume of oil that can be carried. True False Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can represented by the equation \( 3 x^{2}-4 t^{2}=1,000,000 \) (because the builder can return the cut-out squares for credit). The langranian for this constrained optimization is \( L=3 t x^{2}-8 t^{2} x+4 t^{3}+\lambda\left(3 x^{2}-4 t^{2}-1,000,000\right) \). Which of the following represent the first order conditions or a maximum? Check all that apply. \[ \begin{array}{l} \frac{d L}{d x}=6 t x-8 t^{2}+6 x \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d \lambda}=3 x^{2}-4 t^{2}-1,000,000=0 \\ \frac{d L}{d t}=3 x^{2}-8 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}=0 \\ \frac{d L}{d x}=6 t x-8 t^{2}=0 \\ \frac{d L}{d x}=3 t x-8 t^{2} x+6 x \lambda=0 \end{array} \] Once the sides have been folded up and welded in place, the structure is now resembles a tray with no top. This tray has a height of \( t \), with the dimensions of the base \( \frac{(x-2 t) \text { by }(3 x-2 t)}{}- \), which of the following formulas describes the volume of the resulting tray-like structure. For a given value of \( x \), the value of \( t \) that maximizes the volume is \( t^{*}= \) True or False: There is a value of \( x \) that maximizes the volume of oil that can be carried. True False Suppose that a shipbuilder is constrained to use only 1,000,000 square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation \( 3 x^{2}-4 t^{2}=1,000,000 \) (because the builder can return the cut-out squares for credit). The langranian for this constrained optimization is \( L=3 t x^{2}-8 t^{2} x+4 t^{3}+\lambda\left(3 x^{2}-4 t^{2}-1,000,000\right) \). Which of the following represent the first order conditions or a maximum? Check all that apply. \[ \frac{d L}{d x}=6 t x-8 t^{2}+6 x \lambda=0 \] \[ \begin{array}{l} \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d \lambda}=3 x^{2}-4 t^{2}-1,000,000=0 \\ \frac{d L}{d t}=3 x^{2}-8 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}=0 \end{array} \] \[ \begin{array}{l} \frac{d L}{d x}=6 t x-8 t^{2}=0 \\ \frac{d L}{d x}=3 t x-8 t^{2} x+6 x \lambda=0 \end{array} \] Once the sides have been folded up and welded in place, the structure is now resembles a tray with no top. This tray has a height of \( t \), with the dimensions of the base \( (x-2 t) \) by \( (3 x-2 t) \nabla \), which of the following formulas describes the volume of the resulting tray-like structure. \[ \begin{array}{l} V=t(x-t)(3 x-t)=3 t x^{2}-4 t^{2} x+t^{3} \\ V=t(x+2 t)(3 x+t)=3 t x^{2}+7 t^{2} x+2 t^{3} \\ V=t(x-2 t)(3 x-2 t)=3 t x^{2}-8 t^{2} x+4 t^{3} \end{array} \] For a given value of \( x \), the value of \( t \) that maximizes the volume is \( t^{*}= \) Suppose that a shipbuilder is constrained to use only \( 1,000,000 \) square feet of steel sheet to construct an oil tanker. This constraint can be represented by the equation \( 3 x^{2}-4 t^{2}=1,000,000 \) (because the builder can return the cut-out squares for credit). The langranian for this constrained optimization is \( L=3 t x^{2}-8 t^{2} x+4 t^{3}+\lambda\left(3 x^{2}-4 t^{2}-1,000,000\right) \). Which of the following represent the first order conditions or a maximum? Check all that apply. \[ \begin{array}{l} \frac{d L}{d x}=6 t x-8 t^{2}+6 x \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d \lambda}=3 x^{2}-4 t^{2}-1,000,000=0 \\ \frac{d L}{d t}=3 x^{2}-8 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}=0 \\ \frac{d L}{d x}=6 t x-8 t^{2}=0 \\ \frac{d L}{d x}=3 t x-8 t^{2} x+6 x \lambda=0 \end{array} \] The langranian for this constrained optimization is \( L=3 t x^{2}-8 t^{2} x+4 t^{3}+\lambda\left(3 x^{2}-4 t^{2}-1,000,000\right) \). Which of the following represent the first order conditions or a maximum? Check all that apply. \[ \begin{array}{l} \frac{d L}{d x}=6 t x-8 t^{2}+6 x \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d \lambda}=3 x^{2}-4 t^{2}-1,000,000=0 \\ \frac{d L}{d t}=3 x^{2}-8 x t+12 t^{2}-8 t \lambda=0 \\ \frac{d L}{d t}=3 x^{2}-16 x t+12 t^{2}=0 \\ \frac{d L}{d x}=6 t x-8 t^{2}=0 \\ \frac{d L}{d x}=3 t x-8 t^{2} x+6 x \lambda=0 \\ \frac{d L}{d \lambda}=3 x^{2}-4 t^{2}+1,000,000=0 \end{array} \] maximization problem, implying that the solutions for the volume maximizing \( x \) and \( t \) are the same. True False