A technical company is designing a rectangular solar panel installation on a roof using $\displaystyle 300$ metres of boundary material. The design includes a partition running parallel to one of the sides dividing the area (roof) into two sections. Let the length of the side perpendicular to the partition be $\displaystyle x$ metres and with parallel to the partition be y metres. * $\displaystyle \begin{array}{r}\text { 口广 } \\ \text { sinc }\end{array}$ Based on this information, answer the following questions :

(i)

Write the equation for the total boundary material used in the boundary and parallel to the partition in terms of $\displaystyle x$ and $\displaystyle y$.

(ii)

Write the area of the solar panel as a function of $\displaystyle x$.

(iii)

(a)OR(b)

Find the critical points of the area function. Use second derivative test to determine critical points at the maximum area. Also, find the maximum area.

For critical points, put \(\displaystyle \frac{d A}{d x}=0 \Rightarrow x=75\) Also, \(\displaystyle \frac{\boldsymbol{d}^{\mathbf{2}} \boldsymbol{A}}{\boldsymbol{d} \boldsymbol{x}^{\mathbf{2}}}=-\frac{\mathbf{4}}{\mathbf{3}}<\mathbf{0}\). So, \(\displaystyle \boldsymbol{A}\) is maximum at \(\displaystyle \boldsymbol{x}=\mathbf{7 5}\) Also, maximum area is \(\displaystyle \boldsymbol{A}=\frac{\mathbf{7 5}}{\mathbf{3}}(\mathbf{3 0 0}-\mathbf{1 5 0})=\mathbf{3 7 5 0 ~ m}^{\mathbf{2}}\) \[\begin{aligned} & \text { (iii)(b)A= } \frac{x}{3}(300-2 x)=\frac{1}{3}\left(300 x-2 x^{2}\right) \\ & \Rightarrow \frac{d A}{d x}=\frac{1}{3}(300-4 x) \end{aligned} \] For critical points, put \(\displaystyle \frac{d A}{d x}=0 \Rightarrow x=75\) As \(\displaystyle \frac{\boldsymbol{d A}}{\boldsymbol{d} \boldsymbol{x}}\) changes its sign from positive to negative as x passes through \(\displaystyle \boldsymbol{x}=\mathbf{7 5}\) from left to right, whichmeans \(\displaystyle \boldsymbol{x}=\mathbf{7 5}\) is the point of maximum. Also, maximum area is \(\displaystyle \boldsymbol{A}=\frac{\mathbf{7 5}}{\mathbf{3}}(\mathbf{3 0 0}-\mathbf{1 5 0})=\mathbf{3 7 5 0} \mathbf{m}^{\mathbf{2}}\)\(\displaystyle 2 x+2 y=300\) or \(\displaystyle 2 x+4 y=300\) or \(\displaystyle 4 x+4 y=300\) or \(\displaystyle 4 x+3 y=300\) Thesolutions of sub-parts will differ and marks maybe given accordingly.