A bridge is built in the shape of a semi elliptical arch. It has a span of 96 feet. The height of the arch 29 feet from the center is to be 10 feet. Find the height of the arch at its center.
the semi-axis along the base is 48, so we have
x^2/48^2 + y^2/b^2 = 1
The curve contains (29,10), so
29^2/48^2 + 10^2/b^2 = 1
b = 12.55
An archway that is above a highway is in the shape of semi ellipse 60ft wide and 25 ft high. How high is the archway 10 ft away from the center?
To find the height of the arch at its center, we can use the formula for the equation of a semi-ellipse:
y = b * sqrt(1 - (x^2 / a^2))
Where:
- x is the horizontal distance from the center
- y is the vertical distance from the center
- a is the distance from the center to the end of the semi-ellipse (span / 2)
- b is the height of the arch at a distance of 29 feet
Given:
- Span = 96 feet
- Height at 29 feet = 10 feet
First, we find the value of b using the given information:
b = height at 29 feet = 10 feet
Then, we find the value of a:
a = span / 2 = 96 / 2 = 48 feet
Now, we can substitute the values into the equation to find the height at the center (x = 0):
y = b * sqrt(1 - (x^2 / a^2))
y = 10 * sqrt(1 - (0^2 / 48^2))
y = 10 * sqrt(1 - 0)
y = 10 * sqrt(1)
y = 10 * 1
y = 10 feet
Therefore, the height of the arch at its center is 10 feet.
To find the height of the arch at its center, we can use the equation of a semi-ellipse.
The equation for a vertical semi-ellipse can be written as:
(x – h)^2 / a^2 + (y – k)^2 / b^2 = 1
In this case, the semi-ellipse represents the shape of the bridge arch. The center of the semi-ellipse is at the point (h, k), and a and b represent the horizontal and vertical semi-major and semi-minor axes, respectively.
Since the bridge has a span of 96 feet, which is the horizontal diameter of the semi-ellipse, we can conclude that 2a = 96, and thus a = 48.
The height of the arch 29 feet from the center is given as 10 feet. This point lies on the semi-ellipse. Let's call this point (x1, y1), where x1 = 29. Plugging these values into the equation of the semi-ellipse, we get:
(29 – h)^2 / (48)^2 + (10 – k)^2 / b^2 = 1
Now, we can find the value of b by solving for it using the given point on the semi-ellipse. Rearranging the equation, we have:
b^2 = [(29 – h)^2 / (48)^2] / [1 – (10 – k)^2 / b^2]
Multiplying both sides by b^2 and rearranging, we get:
b^4 - b^2 * (10 - k)^2 = (29 - h)^2 / (48)^2 * b^2
b^4 - b^2 * (10 - k)^2 - (29 - h)^2 / (48)^2 * b^2 = 0
This is a quadratic equation in terms of b^2. Solving it will give us two possible values for b^2. However, we can eliminate the negative value since b represents a length, which cannot be negative.
Once we have the value of b, we can find the height of the arch at its center, which is the value of k, by substituting the known values back into the semi-ellipse equation.
Please provide the values of h and k in order to solve the equation and find the height of the arch at its center.