Final answer:
Berri and Alana should stand approximately 74.34 feet apart, which is the distance between the two foci of the semi-elliptical planetarium.
Step-by-step explanation:
The question involves determining how many feet apart Berri and Alana should stand while they are on opposite sides of a planetarium. The planetarium has a dome ceiling in the shape of a semi-ellipse, which is 10 feet high and 75 feet wide. The distance apart they need to stand is the length between the two foci of the ellipse.
To solve this, we use the properties of an ellipse. The major axis is the widest part of the ellipse, which in this case is the width of the planetarium at 75 feet. The major axis is 2a, so a is 37.5 feet. The height of the dome is the minor axis, which is 10 feet, making b, half the minor axis, 5 feet. The distance between the foci (2c) of an ellipse is determined by the equation c^2 = a^2 - b^2. Once c is found, the distance between the foci is 2c, which will give us the distance between Berri and Alana.
Let's calculate: c^2 = 37.5^2 - 5^2 -> c^2 = 1406.25 - 25 -> c = √1381.25. Therefore, c is approximately 37.17 feet. The distance between the foci is 2c, which is roughly 74.34 feet.