# [SOLVED]maximum and minimum 283: minimize cost

#### leprofece

##### Member
283) you want to build a volume "V" shaped geometric body torque limited cylindrical half-spheres. If the material lower semisphere costs twice as much as the material of the sides, and the material of the upper hemisphere costs three times, calculate the dimensions of the body more economic.

answer H = 6R and R = cubic root of ( 3V/22pi)

Ok
V = 2pir2H+4/3pir3

The cost Function where i think it is my mistake is

C = 4pir2H +4pir3

#### MarkFL

Staff member
Re: max and minimun 283

I have no idea what you mean by " 'V' shaped geometric body torque limited cylindrical half-spheres." Is this in fact just a right circular cylinder capped at both ends by hemispheres? If so, you are making the same mistake you made in the boiler problem concerning the volume of this body.

I would let the cost per unit square of the sides of the cylindrical portion be $k$ (in whatever units of currency you choose). Then your cost function would be:

Total cost = k(area of cylindrical portion) + 2k(area of lower hemisphere) + 3k(area of upper hemisphere)

You should now be able to state your objective function and constraint. What do you find?

#### leprofece

##### Member
Re: max and minimun 283

I have no idea what you mean by " 'V' shaped geometric body torque limited cylindrical half-spheres." Is this in fact just a right circular cylinder capped at both ends by hemispheres? If so, you are making the same mistake you made in the boiler problem concerning the volume of this body.

I would let the cost per unit square of the sides of the cylindrical portion be $k$ (in whatever units of currency you choose). Then your cost function would be:

Total cost = k(area of cylindrical portion) + 2k(area of lower hemisphere) + 3k(area of upper hemisphere)

You should now be able to state your objective function and constraint. What do you find?
Cost = k(4pir2·+2k(2/3pir3)+3k 2k(2/3pir3)
This must be the cost equation

#### MarkFL

Staff member
Re: max and minimun 283

Cost = k(4pir2·+2k(2/3pir3)+3k 2k(2/3pir3)
This must be the cost equation
No, what is the formula for the lateral surface of a cylinder? And why does your third term have factors of $2k$ and $3k$? Look at the formula I gave and carefully insert the correct area formulas.

#### leprofece

##### Member
Re: max and minimun 283

No, what is the formula for the lateral surface of a cylinder? And why does your third term have factors of $2k$ and $3k$? Look at the formula I gave and carefully insert the correct area formulas.

C= 2pirh+ 2pir2 +2pir2
This?????

#### MarkFL

Staff member
No. You have the correct lateral surface area of a cylinder, and the correct surface areas of the hemispheres, but you are not taking into account the differing costs per unit area. You want:

$$\displaystyle C(r,h)=k(2\pi rh)+2k\left(2\pi r^2 \right)+3k\left(2\pi r^2 \right)=2k\pi r\left(h+5r \right)$$

Now, can you state the constraint, and solve it for $h$ so that you will be able to express the objective function in terms of just the variable $r$?

#### leprofece

##### Member
No. You have the correct lateral surface area of a cylinder, and the correct surface areas of the hemispheres, but you are not taking into account the differing costs per unit area. You want:

$$\displaystyle C(r,h)=k(2\pi rh)+2k\left(2\pi r^2 \right)+3k\left(2\pi r^2 \right)=2k\pi r\left(h+5r \right)$$

Now, can you state the constraint, and solve it for $h$ so that you will be able to express the objective function in terms of just the variable $r$?
Ok is it ????
V = 2pir2H+4/3pir3

#### MarkFL

Staff member
Ok is it ????
V = 2pir2H+4/3pir3
If I am reading this correctly as:

$$\displaystyle V=2\pi r^2h+\frac{4}{3}\pi r^3$$

then no, that does not described the volume of the body. You have made this same error three times now.

You want:

$$\displaystyle V=\pi r^2h+\frac{4}{3}\pi r^3$$

Recall that the volume of a cylinder is $\pi r^2h$. Now, solve this constraint for $h$ and then substitute for $h$ into the cost function, and then minimize.