If a pebble is dropped into a pond in the shape of an ellipse at the location of one focus, the waves will converge at the location of the other focus. If the pond has a major axis of 34 feet and a minor axis of 16 feet, how far apart are the foci?
To find the distance between the foci of an ellipse when the major and minor axes are known, we can use a formula called the focal length or the distance between foci formula.
Here's how we can apply the formula to solve the problem:
1. Recall that the eccentricity (e) of an ellipse is a measure of how "elongated" it is.
2. Calculate the eccentricity (e) of the ellipse using the formula e = √(a² - b²) / a, where "a" is the length of the major axis and "b" is the length of the minor axis.
In this case, a = 34 feet (the length of the major axis) and b = 16 feet (the length of the minor axis).
e = √(34² - 16²) / 34
e = √(1156 - 256) / 34
e = √900 / 34
e = 30 / 34
e ≈ 0.8824
3. Once you have the eccentricity, you can find the distance between the foci (2c) using the formula 2c = 2ae, where "a" is the length of the major axis and "e" is the eccentricity.
In this case, a = 34 feet (the length of the major axis) and e ≈ 0.8824.
2c = 2(34) * 0.8824
2c = 68 * 0.8824
2c ≈ 60.0696
Therefore, the foci are approximately 60.0696 feet apart.
Better review the properties of an ellipse.
You have a=17 and b=8
Now, each focus is at a distance c from the center, where
b^2 + c^2 = a^2
(recall your basic Pythagorean triples here, to save time)