Activity III: The Blue Room in the White House (Grades 6-10)
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Comparative Costs: The Early 19th Century and Today
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The Blue Room in the White House |
More Math Concepts
- Understand the definition of an ellipse and how to produce a graph of an ellipse
- Find the equation of an ellipse using the endpoints of the major and minor axes
- Find the coordinates of the foci and the eccentricity of an ellipse
Standard 2: Patterns, Functions: use symbolic forms to represent and analyze mathematical situations and structures.
Standard 3: Geometry and Spatial Sense: select and use different representational systems including coordinate geometry.
One of the most unusual rooms in the president's residence, the White House, is the Blue Room. Unlike most rooms, the Blue Room is elliptical or egg shaped. (See Figure 1.)
Figure 1. The Blue Room.
A history of the Blue Room as well as pictures can be found at:
The Blue Room was designed by architect James Hoban. Rooms in most homes (the president's or yours) are rectangular with adjoining walls forming right angles with each other.
1. Why do most architects or home designers use right angles?
2. Suppose you wanted to use some other angle (say, 45 degrees or 120 degrees) for corners to meet in a home. Describe any problems that might arise.
Hoban chose to make the Blue Room 39 feet and ¾" long and 29 feet and 4 ¾" wide.
3. Use the graph in Figure 2 to sketch an ellipse similar to the Blue Room floor plan.
For the remainder of this activity, approximate the length of the Blue Room to be 40 feet and the width to be 30 feet.
Figure 2. Graph for your drawing of the Blue Room.
The longest part of an ellipse is called the major axis. The short side is the minor axis.
4. What is the length of the major axis of the Blue Room? The minor axis?
The equation of an ellipse is where a is half the major axis and b is half the minor axis.
5. What is the equation for the ellipse traced by the Blue Room?
An ellipse is formed by identifying two focal points (sometimes called foci). Formally, an ellipse is the set of points whose distance from the two foci is always a constant (a constant bigger than the distance between the two foci). That's hard to understand but easy to see from a picture. Imagine two foci (the dots in the picture in Figure 3) and a string attached between them (the string is longer than the distance between the two foci). Put a pencil on the string and stretch the string taut. Now trace.
Figure 3. Foci with string.
You might want to try this outdoors by placing two pegs in the ground and tying some rope between them. Use a stick or chalk to trace the ellipse. You might also construct your own Ellipse Drawing Board as described at http://nths.newtrier.k12.il.us/academics/math/Connections/curves/elldrabd.htm.
The coordinates for the foci can be found using the coordinates for the major and minor axes. If (-a, 0) and ( , 0) are the endpoints of the major axis and (0, -b) and (0, b) are the endpoints of the minor axis, then the focal points can be found using a2 = b2 + c2. This makes , and the two foci are (-c, 0) and (c, 0).
6. Find the coordinates of the foci for the Blue Room.
If you plot the foci on the coordinate system in Figure 2 above, you can place a piece of string with each end at a foci and the middle stretched taut with a pencil ending on the ellipse. Now, as you move the pencil with the string taut, you can trace the ellipse.
The eccentricity of an ellipse is a measure of how flat or round it is. A circle is really an ellipse with the foci sitting one on top of the other. Its eccentricity is 1. The eccentricity of an ellipse is defined to be e = c/a (where c is the x-coordinate of the focus and a is the x-coordinate of the major axis).
7.What is the eccentricity of the Blue Room?