A swimming pool in the shape of an ellipse is modeled by x^2/324+y^2/196=1, where x and y are measured in feet.
Find the shortest distance (in feet) across the center of the pool?
i think its 28 feet....
2b = length of the minor axis
2 * 14 = 28
Good job!
Thank you very much!!
To find the shortest distance across the center of the pool, we need to find the length of the major axis of the ellipse.
In the equation of the ellipse, x^2/324 + y^2/196 = 1, we can see that the denominators under x^2 and y^2 are the squares of their respective semi-major axes.
Therefore, the semi-major axis in the x-direction is 18 feet (sqrt(324)) and the semi-major axis in the y-direction is 14 feet (sqrt(196)).
Since we are looking for the distance across the center of the pool, it is equivalent to two times the semi-major axis in the y-direction.
Thus, the shortest distance across the center of the pool is 2 * 14 feet = 28 feet.
Therefore, your answer of 28 feet is correct.
To find the shortest distance across the center of the pool, we need to find the distance between the points where the ellipse intersects its major axis.
The equation of the ellipse is given by x^2/324 + y^2/196 = 1.
To find the points where the ellipse intersects the major axis, we set y = 0 and solve for x.
Plugging y = 0 into the equation, we get x^2/324 = 1.
Multiplying both sides by 324 gives us x^2 = 324.
Taking the square root of both sides, we get x = ±√324 = ±18.
Therefore, the points where the ellipse intersects the major axis are (-18, 0) and (18, 0).
The distance between these two points is given by the distance formula:
distance = √((x2 - x1)^2 + (y2 - y1)^2)
Let's substitute the values into the formula:
distance = √((18 - (-18))^2 + (0 - 0)^2)
Simplifying, we get:
distance = √((18 + 18)^2 + (0 - 0)^2)
distance = √(36^2 + 0^2)
distance = √(1296 + 0)
distance = √1296
distance = 36 feet
Therefore, the shortest distance across the center of the pool is 36 feet, not 28 feet.