Problem with differentiation and area of components

[TW Cantor]

Homework Statement

The diagram attached shows a solid engineering component which consists of a solid cylinder of length L and radius r, together with a right circular cone with semi-vertical angle = 11 degrees.
The total surface area of the component is 481.
The component is to be manufactured so as to have maximum volume.

Calculate the values of L and R for maximum volume.

Homework Equations

Surface area of cylinder = 2*pi*radius*length + 2*pi*radius²
Surface area of cone = pi*radius*length of slope + pi*radius²
Volume of cylinder = pi*radius²*length
Volume of cone = (pi*radius²*heigth)/3

The Attempt at a Solution

i guess i have to get an equation for the total volume in terms of r but i can't seem to think of a way to find the radius in terms of L. i would then differentiate this equation and equal it to 0 to find the maximum volume

 

[LCKurtz]

Homework Statement

The diagram attached shows a solid engineering component which consists of a solid cylinder of length L and radius r, together with a right circular cone with semi-vertical angle = 11 degrees.
The total surface area of the component is 481.
The component is to be manufactured so as to have maximum volume.

Calculate the values of L and R for maximum volume.

Homework Equations

Surface area of cylinder = 2*pi*radius*length + 2*pi*radius²

When the parts are joined, the base of the cone and the end of the cylinder where it is attached are inside the solid. They wouldn't be included in the surface area. So don't count both ends of the cone.

Surface area of cone = pi*radius*length of slope + pi*radius²

And don't count the base of the cone in the surface area.

Volume of cylinder = pi*radius²*length
Volume of cone = (pi*radius²*heigth)/3

The Attempt at a Solution

i guess i have to get an equation for the total volume in terms of r but i can't seem to think of a way to find the radius in terms of L. i would then differentiate this equation and equal it to 0 to find the maximum volume

First you need to correct your equations and write them in terms of r and L. Then when you calculate the total surface area and set it equal to 481, you should be able to solve it for L in terms of r. And once you have r you can get L.

 

[TW Cantor]

hi LCKurtz :-)

so I've worked out that L in terms of r is equal to:

L=(π*r²+((π*r²)/sin(11))-481)/(2*π*r)

i guess i then have to get combine the two volume equations for a cone and a cylinder to get total volume in terms of r. i then differentiate the total volume and equal it to 0 to find r at maximum volume?

 

[LCKurtz]

hi LCKurtz :-)

so I've worked out that L in terms of r is equal to:

L=(π*r²+((π*r²)/sin(11))-481)/(2*π*r)

i guess i then have to get combine the two volume equations for a cone and a cylinder to get total volume in terms of r. i then differentiate the total volume and equal it to 0 to find r at maximum volume?

I think you need (481 - expression) instead of (expression - 481) in the numerator, so check your sign. Other than that, yes you have the right plan of what to do next.

 

[TW Cantor]

yes i have now completed this problem :-)
thanks a lot for your hints ;-)