Answer:
[tex]18.73ft^3[/tex]
Step-by-step explanation:
Let
Side of square base=x
Height of rectangular box=y
Area of square base=Area of top=[tex](side)^2=x^2[/tex]
Area of one side face=[tex]l\times b=xy[/tex]
Cost of bottom=$9 per square ft
Cost of top=$5 square ft
Cost of sides=$4 per square ft
Total cost=$204
Volume of rectangular box=[tex]V=lbh=x^2y[/tex]
Total cost=[tex]9(x^2)+5x^2+4(4xy)=14x^2+16xy[/tex]
[tex]204=14x^2+16xy[/tex]
[tex]204-14x^2=16xy[/tex]
[tex]y=\frac{204-14x^2}{16x}=\frac{102-7x^2}{8x}[/tex]
Substitute the values of y
[tex]V(x)=x^2\times \frac{102-7x^2}{8x}=\frac{1}{8}(102x-7x^3)[/tex]
Differentiate w.r.t x
[tex]V'(x)=\frac{1}{8}(102-21x^2)=0[/tex]
[tex]V'(x)=0[/tex]
[tex]\frac{1}{8}(102-21x^2)=0[/tex]
[tex]102-21x^2=0[/tex]
[tex]102=21x^2[/tex]
[tex]x^2=\frac{102}{21}=4.85[/tex]
[tex]x=\sqrt{4.85}=2.2[/tex]
It takes positive because side length cannot be negative.
Again differentiate w.r. t x
[tex]V''(x)=\frac{1}{8}(-42x)[/tex]
Substitute the value
[tex]V''(2.2)=-\frac{42}{8}(2.2)=-11.55<0[/tex]
Hence, the volume of box is maximum at x=2.2 ft
Substitute the value of x
[tex]y=\frac{102-7(2.2)^2}{8(2.2)}=3.87 ft[/tex]
Greatest volume of box=[tex]x^2y=(2.2)^2\times 3.87=18.73 ft^3[/tex]
