An arch is in the form of a semi-ellipse, with its major axis as the span. If the span is 80 feet and the height is 30 feet write the standard equation.
To write the standard equation of a semi-ellipse, we need the equation in the form:
((x - h)² / a²) + ((y - k)² / b²) = 1
where (h, k) represents the coordinates of the center of the ellipse, a is the semi-major axis, and b is the semi-minor axis.
In this case, the major axis of the semi-ellipse is the span, which is 80 feet. The height is given as 30 feet.
First, we need to find the coordinates of the center, (h, k), which is the midpoint of the span. Since the span is along the x-axis, the center will have the x-coordinate of the midpoint of the span and the y-coordinate will be the height.
The x-coordinate of the center is:
h = (0 + 80) / 2 = 40
The y-coordinate of the center is:
k = 30
Next, we need to find the semi-major axis, which is half the length of the span:
a = 80 / 2 = 40
The semi-minor axis, b, is half the height:
b = 30 / 2 = 15
Now, we can write the standard equation using the values we obtained:
((x - 40)² / 40²) + ((y - 30)² / 15²) = 1
So, the standard equation of the given semi-ellipse is ((x - 40)² / 1600) + ((y - 30)² / 225) = 1.
To find the standard equation of a semi-ellipse, we need to have the coordinates of the center (h, k), the lengths of the major and minor axes (2a and 2b), and the orientation of the semi-ellipse.
In this case, the major axis is the span, which has a length of 80 feet, and the height is the distance between the center and the foci, which is half of the minor axis, giving us a length of 30 feet. Since the major axis is horizontal, the semi-ellipse is oriented along the x-axis.
First, let's find the center of the semi-ellipse. The x-coordinate of the center is simply the midpoint of the major axis, which is (80/2) = 40. The y-coordinate of the center is 0 since the semi-ellipse is oriented along the x-axis.
So, the center of the semi-ellipse is (40, 0). The major axis (2a) has a length of 80 feet, so a = 40. The height (30 feet) is half of the minor axis, so b = 30.
The equation of a semi-ellipse with its center at (h, k) is:
(x-h)^2/a^2 + (y-k)^2/b^2 = 1
Plugging in the values, we have:
(x-40)^2/(40)^2 + (y-0)^2/(30)^2 = 1
Simplifying, we get:
(x-40)^2/1600 + y^2/900 = 1
Therefore, the standard equation of the semi-ellipse is:
(x-40)^2/1600 + y^2/900 = 1
X^(2)/40^(2) + y^(2)/30^(2)
I think that's the answer...😊☺😅😅😴
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