Answer:
a.[tex]6x^2+\frac{768}{x}=6x^2+\frac{768}{x}[/tex]
b. 288 sq units
Step-by-step explanation:
Given the dimensions of the base sides and the cuboids volume, we can calculate its height:
[tex]v=lwh\\\\288=3x(x)\times h\\\\288=3x^2h\\\\h=\frac{288}{3x^2}=\frac{96}{x^2}[/tex]
Having determined h=[tex]\frac{96}{x^2}[/tex].
The surface area of the cuboid is the sum of all its faces area;
[tex]A=2lw+2lh+2hw\\\\=2(3x\times x)+2(3x\times \frac{96}{x^2})+2(x\times\frac{96}{x^2})\\\\=6x^2+\frac{576}{x}+\frac{192}{x}\\\\=6x^2+\frac{768}{x}[/tex]
[tex]6x^2+\frac{768}{x}=6x^2+\frac{768}{x}[/tex]=A, hence, proved!
b. Find stationary value of A
We find the critical point of the function:
[tex]f\prime(x)=6x^2+\frac{768}{x}, x<0,x>0\\\\x=0\\\\x=(\frac{128}{2})^{1/3}\\\\x=4[/tex]
Hence, x is undefined. The stationary area is therefore calculated as:
[tex]A=6x^2+768/x\\\\=6(4^2)+768/4=288[/tex]
The area is 288 sq units
