An arch has the shape of a semi-ellipse (the top half of an...

An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a
span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of
the arch at a distance of 4 feet from the center.

Key Concepts

Substitution and Evaluation

Once the equation of the ellipse is established, substituting specific values (like a given x-coordinate) into the equation enables one to solve for the corresponding y-coordinate. This process is essential in determining particular dimensions or positions on the ellipse, which is crucial for applications such as calculating the height of an arch at a specific distance from its center.

Ellipse

An ellipse is a type of conic section defined as the set of all points for which the sum of the distances to two fixed points (the foci) is constant. In coordinate geometry, it can be represented in a standard form that helps in analyzing its properties such as its axes, center, and the lengths of its semi-major and semi-minor axes.

Standard Form Equation

The standard equation of an ellipse centered at the origin is written as (x²/a²) + (y²/b²) = 1, where 'a' represents the semi-axis length in the x-direction and 'b' represents the semi-axis length in the y-direction. This form allows for incorporating given dimensions of width (span) and height directly into the equation.

Semi-Ellipse

A semi-ellipse is the portion of an ellipse cut by a line, often representing just the upper or lower half. In many practical applications, such as architectural arches, only one half of the ellipse is used. This concept involves using the relevant half of the ellipse equation to model physical structures.

Transcript

00:02 So we have a word problem that says we have an arc the shape of a semi -elipse.

00:09 So a semi -elipse is just the top half of an ellipse.

00:13 So i'm going to draw it on a graph so it's easier to see.

00:17 So semi -look something like that.

00:24 We know that it has a height of eight feet.

00:28 So this right here, this whole length here would be eight feet.

00:34 And we also know that it has a span of 20 feet.

00:37 So when it says span of 20 feet, that means the whole length is 20 feet.

00:43 Therefore, this part here is half of that, so it's 10.

00:51 So we want to find the equation of an ellipse and use that to find the height at a distance of 4 feet from the center.

01:00 So 4 feet from the center would probably be like around here, and we want to find the height at this point.

01:07 So this is a point on the ellipse, so the coordinate will be 4 because we'd probably be like around here, and we want to find the height at this point.

01:12 Know this distance is 4.

01:14 However, we want to look for the height, which would just call y.

01:19 So we need to find the equation.

01:22 So we know that our height, this will be b, and our length is 10, which would be a.

01:30 And now all we have to do is plug all these into our standard equation.

01:41 So the standard equation is x squared over a squared plus y squared over b squared equals 1...

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