Lesson 12.3.notebook
12.3 Quadric Surfaces and Inscribed Figures Day 1
March 07, 2012
Lesson 12.3.notebook
March 07, 2012
Lesson 12.3.notebook
March 07, 2012
Lesson 12.3
I'm not going to vacuum
'til Sears makes one you can ride on.
­Roseanne Barr
Lesson 12.3.notebook
March 07, 2012
Ellipse Parametric Function : Here are the parametric equations for a particular ellipse :
x = 1 + 5 cos t
y = 2 + 3 sin t
A. With your calculator in parametric mode, plot this parametric function.
B. From the graph, where does the center seem to be ? What do the x­radius and y­radius seem to be ?
C. How can you identify the center and radii from the parametric equations ?
D. Eliminate the parameter and write the equation of the ellipse in standard form.
1. Exploration 12.3a
Lesson 12.3.notebook
March 07, 2012
Lesson 12.3.notebook
March 07, 2012
2. Definition: Quadric Surface
Three dimensional figure of a conic section,
in three variables x, y, and z.
Lesson 12.3.notebook
March 07, 2012
Lesson 12.3.notebook
3. Definition: Prolate spheroid:
An ellipsoid that's created by rotations around the major axis of the ellipse.
4. Definition: Oblate Spheroid: An ellipsoid that's created by rotations around the minor axis of the ellipse.
Exploration 12.3b
March 07, 2012
Lesson 12.3.notebook
March 07, 2012
Assignment: p. 562 QR 1 – 10, 1 – 10 Submarine Problem : The submarine pressure hull can be built in the shape of a frustum of a cone. A frustum of a cone is a cone with its vertex cut off. The larger base of the frustum is the base of the half ellipsoid. The smaller base of the frustum touches the half ellipsoid at the sample point (x,y). .
(x,y)
225 x 2 + 900 y 2 = 202,500
Lesson 12.3.notebook
March 07, 2012
A. Prove the volume of the frustum of a
cone is V = 1/3 π h ( R 2 +Rr + r 2)
where h is the altitude of the frustum ( between the two circular bases)
and R and r are the radii of the two bases.
B. Find the maximum volume the frustum can have.
Ventilator Duct Problem : A sheet metal duct connects a 20" by 60" vertical rectangular opening in one wall to a 50" by 30" vertical rectangular opening in a wall 100" away. A cross section is taken through the duct, parallel to the two rectangular openings, at a point x" from the left wall. The lengths y and z of the cross section are linear functions of x.
100
20
y
60
z
30
50
Lesson 12.3.notebook
March 07, 2012
A. Write equations expressing y and z as functions of x.
B. Find the area of the cross section as a function of x.
C. At what values of x do the maximum and
minimum cross­sectional areas occur ? What are these areas ?
Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 ft. wide is 0.4 foot higher in the center than it is on the sides.
A. Find an equation of the parabola that models the
road surface.
B. How far from the center of the road is the road
surface 0.1 foot lower than in the middle ?
Lesson 12.3.notebook
March 07, 2012
Find the points on the ellipse 4 x 2 + y 2 = 4
that are the farthest away from the point ( 1 , 0 ).
Find the dimensions of the rectangle of largest area that has its base on the x­axis and its other two vertices above the x­axis and lying on the parabola y = 8 ­ x 2.
Lesson 12.3.notebook
The part of the parabola y = 4 ­ x 2 from x = 0 to x = 2 is rotated about the y­ axis to form a surface. A cone is inscribed in the resulting paraboloid with its vertex at the origin and its base touching the parabola. At what radius and altitude does the maximum volume occur ? What is this maximum volume ?
March 07, 2012