# Math Help - Calc BC optimization problems

1. ## Calc BC optimization problems

Okay so i've spent easily over an hour trying to figure out these questions >_< I'd really appreciate some help with these. I just don't kno wwhere to start.

The whole problems are as follows:

1) A right circular cone has base radius 5 and altitude 12. A cylinder is to be inscribed int he cone so that the axis of the cylinder coincides with the axis of the cone. Given that the radius of the cylinder must be between 2 and 4 inclusive, find the value of the radius for which the lateral surface area of the cylinder is a maximum. Justify your answer. (NOTE:The lateral surface area of the cylinder does NOT include the bases.)

2) "A piece of wire 60 inches long is cut into six sections, two of one length and four of another length. Each of the two sections having the same length is bent into the form of a circle and the two circles are then joined by the four remaining sections to make a frame for a model of a right circular cylinder, as shown in the accompanying figure. (The picture is a Cone, with 4 pillars and two circles as base and top.)

a) Find the lengths of the sections which will make the cylinder of maximum volume.

Any help would be greatly appreciated ^^

2. Originally Posted by jing
Okay so i've spent easily over an hour trying to figure out these questions >_< I'd really appreciate some help with these. I just don't kno wwhere to start.

The whole problems are as follows:

1) A right circular cone has base radius 5 and altitude 12. A cylinder is to be inscribed int he cone so that the axis of the cylinder coincides with the axis of the cone. Given that the radius of the cylinder must be between 2 and 4 inclusive, find the value of the radius for which the lateral surface area of the cylinder is a maximum. Justify your answer. (NOTE:The lateral surface area of the cylinder does NOT include the bases.)

2) "A piece of wire 60 inches long is cut into six sections, two of one length and four of another length. Each of the two sections having the same length is bent into the form of a circle and the two circles are then joined by the four remaining sections to make a frame for a model of a right circular cylinder, as shown in the accompanying figure. (The picture is a Cone, with 4 pillars and two circles as base and top.)

a) Find the lengths of the sections which will make the cylinder of maximum volume.

Any help would be greatly appreciated ^^
1/
Draw a diagram

let h be height

tan(x)=h/5-r=12/5
h=2.5(5-r)
A=24*pi*(5r-r^2)

Take it from here, find the maximum and set it to 0

3. Originally Posted by jing
Okay so i've spent easily over an hour trying to figure out these questions >_< I'd really appreciate some help with these. I just don't kno wwhere to start.

The whole problems are as follows:

1) A right circular cone has base radius 5 and altitude 12. A cylinder is to be inscribed int he cone so that the axis of the cylinder coincides with the axis of the cone. Given that the radius of the cylinder must be between 2 and 4 inclusive, find the value of the radius for which the lateral surface area of the cylinder is a maximum. Justify your answer. (NOTE:The lateral surface area of the cylinder does NOT include the bases.)

2) "A piece of wire 60 inches long is cut into six sections, two of one length and four of another length. Each of the two sections having the same length is bent into the form of a circle and the two circles are then joined by the four remaining sections to make a frame for a model of a right circular cylinder, as shown in the accompanying figure. (The picture is a Cone, with 4 pillars and two circles as base and top.)

a) Find the lengths of the sections which will make the cylinder of maximum volume.

Any help would be greatly appreciated ^^
2/
V of Cylinder formula=pir^2h

use the constraint and do the algebra;

(1/4)(60-4pir)=15-pir=h

=pir^2(15-pir)

=15pir^2-pi^2r^3

set it to zero

=2(15)(pi)(r)-pi^2(3)(r^2)

=3(pi)(r)(10-pi*r)=0

r=10/pi

now finish up

4. 1) sketch a side 2D view. center the vertex at (0,12), with base on the x-axis from -5 to 5.

slant has equation $y = -\frac{12}{5} x$

inscribe a rectangle with two sides on the coordinate axes and the opposite corner on the slant line in quad I.

lateral surface area of the cylinder is

$A = 2\pi rh$

$A = 2\pi x \cdot \left(-\frac{12}{5} x\right)$

find $\frac{dA}{dx}$ and find the radius "x" that maximizes the surface area.
don't forget about endpoint extrema and the domain for x.

5. Originally Posted by jing
Okay so i've spent easily over an hour trying to figure out these questions >_< I'd really appreciate some help with these. I just don't kno wwhere to start.

The whole problems are as follows:

1) A right circular cone has base radius 5 and altitude 12. A cylinder is to be inscribed int he cone so that the axis of the cylinder coincides with the axis of the cone. Given that the radius of the cylinder must be between 2 and 4 inclusive, find the value of the radius for which the lateral surface area of the cylinder is a maximum. Justify your answer. (NOTE:The lateral surface area of the cylinder does NOT include the bases.)

2) "A piece of wire 60 inches long is cut into six sections, two of one length and four of another length. Each of the two sections having the same length is bent into the form of a circle and the two circles are then joined by the four remaining sections to make a frame for a model of a right circular cylinder, as shown in the accompanying figure. (The picture is a Cone, with 4 pillars and two circles as base and top.)

a) Find the lengths of the sections which will make the cylinder of maximum volume.

Any help would be greatly appreciated ^^
2b/ to justify make sure d^2v/dr^2>0 at 10/pi

hint: check the second derivative

6. thanks alot everybody for your help ^^