Question

A silo is a storage building shaped like a cylinder with half a sphere on top. A farmer has 1000 square feet of material with which to build a silo.

Answer 1

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

`S=4\pi r^(2)+2\pi rh`

By replacing S = 1000

`1000=4\pi r^(2)+2\pi rh`

`2\pi rh=1000-4\pi r^(2)`

`h=(1)/(2 \pi r)[1000-4\pi r^(2)]`

b). Expression representing the volume is given by

`V=(2)/(3)\pi r^(3)+\pi r^(2)h`

By replacing the value of h from option (a) to the option (b).

`V=(2)/(3)\pi r^(3)+\pi r^(2)[((1)/(2\pi r))(1000-4\pi r^(2))]`

`V=(2)/(3)\pi r^(3)+500r-2\pi r^(3)`

`V=500r-(4)/(3)\pi r^(3)`

c). For the maximum volume of silo we will find the derivative of V and equate it to zero.

`V'=500-4\pi r^(2)` = 0

`500=4\pi r^(2)`

`r^(2)=(125)/(\pi )`

`r=\sqrt{(125)/(\pi )}`

`r={√(39.77) }`

r = 6.31 feet

For r = 6.31 ft,

`h=[(500)/(\pi r)-2r]`

`h=[(500)/(\pi (6.31))-2(6.31)]`

`h=[25.21-12.62]`

h = 12.59 ft

Therefore, silo will have 6.31 ft height and 12.59 ft height for the maximum volume.