Question

This is Calculus 1 Linear Optimization Problem! MUST SHOW ALL THE JUSTIFICATION!!!Question 1

Answer 1

Question 1;

`\begin{gathered} FG^2=L^2+Z^2\text{ \lparen By Pythagoras theorem\rparen} \\ \\ FE^2=Y^2+(6-X)^2\text{ \lparen By Pythagoras theorem\rparen} \\ \\ FG^2+FE^2=GE^2\text{ \lparen By Pythagoras theorem\rparen} \\ \\ HD=Y\text{ \lparen Since FHDA is a rectangle\rparen} \\ \\ GE^2=X^2+(Z+Y)^2\text{ \lparen By Pythagoras theorem\rparen} \\ \\ Tying\text{ all 4 equations together, we have:} \\ \\ FG^2+FE^2=GE^2 \\ (L^2+Z^2)+(Y^2+(6-X)^2)=X^2+(Z+Y)^2 \\ L^2+Z^2+Y^2+(6-X)^2=X^2+(Z+Y)^2 \\ \text{ Expand the brackets} \\ L^2+Z^2+Y^2+36-12X+X^2=X^2+Z^2+2ZY+Y^2 \\ \text{ Collect like terms} \\ \\ L^2+Z^2-Z^2+Y^2-Y^2+X^2-X^2-12X-2ZY=-36 \\ L^2-12X-2ZY=-36 \\ L^2=12X+2ZY-36 \\ \\ \therefore L=√(12X+2ZY-36) \end{gathered}`

Final Answer

The expression for L in terms of X,Y, and Z is:

`L=√(12X+2ZY-36)`