step 1
Find out the equation of the volume of the box
[tex]\begin{gathered} V=x^2h \\ V=171,500\text{ cm}^3 \\ 171,500=x^2h \\ h=\frac{171,500}{x^2} \end{gathered}[/tex]step 2
Find out the expression for the surface area
The surface area is given by the expression
[tex]A=x^2+4xh[/tex]substitute the value of h
[tex]\begin{gathered} A=x^2+4x\frac{171,500}{x^2} \\ simplify \\ A(x)=x^2+\frac{686,000}{x} \\ \\ A(x)=\frac{x^3+686,000}{x} \end{gathered}[/tex]step 3
Find out the derivative A'(x)
[tex]A^{\prime}(x)=2x-\frac{686,000}{x^2}[/tex]step 4
Equate the derivative to zero
[tex]\begin{gathered} 2x-\frac{686,000}{x^2}=0 \\ 2x=\frac{686,000}{x^2} \\ \\ 2x^3=686,000 \\ x^3=343,000 \\ x=70 \end{gathered}[/tex]step 5
Find out the second derivative A''(x)
[tex]A^{\prime}^{\prime}(x)=2+\frac{1,372,000}{x^3}[/tex]Evaluate the second derivative for x=70
[tex]\begin{gathered} A^{\prime}^{\prime}(x)=2+\frac{1,372,000}{(70)^3} \\ A^{\prime}^{\prime}(x)\text{ is positive} \\ that\text{ means} \\ The\text{ value of A\lparen x\rparen i}s\text{ a maximum for x=70 cm} \end{gathered}[/tex]