Question

An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river 20 meters wide. The height of the arch is 6 meters above the center of the river. Write an equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch.

Answer 1

Title:

Check the explanation.

Step-by-step explanation:

The standard form of an ellipse is $\frac{x^{2} }{a^{2} } + \frac{y^{2} }{b^{2} } = 1$.

The span of the river is 20 meters. As per the given condition, y-axis will pass through the middle point of the river. That is why, the value of a will be $\frac{20}{2} = 10$.

Since the height of the arc is 6, the value of b is 6.

The equation of the arc will become $\frac{x^2}{{10}^2} + \frac{y^2}{6^2} = 1$.

Answer 2

The equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch is $\frac{x^{2}}{100} + \frac{y^{2}}{36} = 1$, for $x \in [-10, 10]$ and $y\in [0, 6]$.

Let be the arch an upper half of an ellipse centered in the midpoint of the surface line of the river, which is represented by the following formula and conditions:

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$, for $x \in [-a, a]$ and $y\in [0, b]$ (1)

Where:

• $x$ - Horizontal position, in meters.
• $y$ - Vertical position, in meters.
• $a$ - Horizontal semiaxis length.
• $b$ - Vertical semiaxis length.

If we know that $a = 10$ and $b = 6$, then the equation of the arch is:

$\frac{x^{2}}{100} + \frac{y^{2}}{36} = 1$ (2)

The equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch is $\frac{x^{2}}{100} + \frac{y^{2}}{36} = 1$, for $x \in [-10, 10]$ and $y\in [0, 6]$.

To learn more on ellipses, we kindly invite to check this verified question: https://brainly.com/question/19507943