In an ellipse. If the talker stands 15 feet from the wall and the distance from the middle of the room to the listener is 45 feet, then how wide (top to bottom) does the room have to be?
To find the width (top to bottom) of the room in an ellipse, we need to use the formula for the equation of an ellipse. The equation of an ellipse is:
((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1
Where (h,k) is the center of the ellipse, and a and b are the lengths of the horizontal and vertical axes, respectively.
In this case, the talker is standing 15 feet from the wall, so we can assume that the horizontal axis of the ellipse is 30 feet (15 feet on each side of the wall). The listener is 45 feet from the center of the room, so the vertical axis of the ellipse is 90 feet (45 feet on each side of the center).
Using these values, we can substitute them into the equation of the ellipse:
((x-0)^2)/30^2 + ((y-0)^2)/90^2 = 1
Simplifying the equation, we get:
x^2/900 + y^2/8100 = 1
Now we can rearrange the equation to solve for y:
y^2 = 8100 - 8100x^2/900
y^2 = 8100 - 9x^2
To find the width (top to bottom) of the room, we need to find the value of y when x=0. Substituting x=0 into the equation, we get:
y^2 = 8100 - 9(0)^2
y^2 = 8100
Taking the square root of both sides, we get:
y = ±90
Since we are interested in the positive value of y, the width (top to bottom) of the room is 90 feet.