Statuary Hal is an elliptical room in the U.S. Capitol. Teh room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. Statuary Hall is 46 ft. wide and 97 ft. long.

a) Find an equation that models the shape of the room.
b) How far apart are the two foci?
c) What is the area of the floor of the room?

Just verifying if what I have done so far is correct...

a) 2a = 97
a = 48.5

2b = 46
b = 23

x^2/(48.5^2) + y^2/(23^2) = 1

b) c^2 = a^2 - b^2
c^2 = 48.5^2 - 23^2
c = 42.7

42.7 * 2 = 85.4 feet apart.

Please correct me if I am doing anything wrong.

I agree with all your answers, except you forgot to find the area

c) area = πab = π(48.5)(23) = 1115.5π or appr3504.4 square feet

"Statuary Hall is 46 ft. wide and 97 ft. long." That's actually wrong. Instead, "Statuary Hall is 92 ft. wide and 97 ft. long." The hall is only half an ellipse as a set of columns cuts off the other (non-existing) half of the ellipse. Look at floor plans and you can see that the hall is almost circular.

Your calculations for part a) and b) are correct!

For part c) to find the area of the floor of the room, you can use the formula for the area of an ellipse, which is given by A = πab, where a and b are the lengths of the semi-major and semi-minor axes of the ellipse, respectively.

In this case, the semi-major axis (a) is the distance from the center of the room to the shorter side, which is 23 ft.
The semi-minor axis (b) is the distance from the center of the room to the longer side, which is 48.5 ft.

So, the area of the floor of the room can be calculated as:
A = π * 23 * 48.5 ≈ 3531.04 square feet

Therefore, the area of the floor of the room is approximately 3531.04 square feet.
Well done on your calculations so far!

a) You are correct in finding the equation that models the shape of the room. The correct equation is:

x^2/(48.5^2) + y^2/(23^2) = 1

b) However, you made a mistake in calculating the distance between the two foci. The correct formula to find the distance between the foci is: c = √(a^2 - b^2)

Plugging in the correct values:
c = √(48.5^2 - 23^2)
c ≈ √(2352.25 - 529)
c ≈ √1823.25
c ≈ 42.72

So the distance between the two foci is approximately 42.72 feet.

c) To find the area of the floor of the room, you can use the formula for the area of an ellipse, which is A = πab.

Plugging in the correct values:
A = π * 48.5 * 23
A ≈ 1763.21 square feet

So the area of the floor of the room is approximately 1763.21 square feet.

Overall, your calculations for parts a, b, and c are almost correct, with an error in part b.