Answer:
Step-by-step explanation:
The given question is incomplete; please find the complete question enclosed as an attachment.
Area of the material by which silo is to be build = 1000 square ft
a). Since Surface area of the silo is represented by the formula
[tex]S=4\pi r^{2}+2\pi rh[/tex]
By replacing S = 1000
[tex]1000=4\pi r^{2}+2\pi rh[/tex]
[tex]2\pi rh=1000-4\pi r^{2}[/tex]
[tex]h=\frac{1}{2 \pi r}[1000-4\pi r^{2}][/tex]
b). Expression representing the volume is given by
[tex]V=\frac{2}{3}\pi r^{3}+\pi r^{2}h[/tex]
By replacing the value of h from option (a) to the option (b).
[tex]V=\frac{2}{3}\pi r^{3}+\pi r^{2}[(\frac{1}{2\pi r})(1000-4\pi r^{2})][/tex]
[tex]V=\frac{2}{3}\pi r^{3}+500r-2\pi r^{3}[/tex]
[tex]V=500r-\frac{4}{3}\pi r^{3}[/tex]
c). For the maximum volume of silo we will find the derivative of V and equate it to zero.
[tex]V'=500-4\pi r^{2}[/tex] = 0
[tex]500=4\pi r^{2}[/tex]
[tex]r^{2}=\frac{125}{\pi }[/tex]
[tex]r=\sqrt{\frac{125}{\pi }}[/tex]
[tex]r={\sqrt{39.77} }[/tex]
r = 6.31 feet
For r = 6.31 ft,
[tex]h=[\frac{500}{\pi r}-2r][/tex]
[tex]h=[\frac{500}{\pi (6.31)}-2(6.31)][/tex]
[tex]h=[25.21-12.62][/tex]
h = 12.59 ft
Therefore, silo will have 6.31 ft height and 12.59 ft height for the maximum volume.
