The “Hatch Shell” in Boston has the arch with semi-elliptical form with a horizontal major axis. The base of the arch is 110-foot long and the highest part of the arch is 40 feet above the scene. Find the height of the arch of the “Hatch Shell” 33 feet away from the midpoint of the base of the arch.

Draw a diagram, with the center of the base at (0,0)

It is easy to see that the equation of the ellipse is

x^2/55^2 + y^2/40^2 = 1

Now just find y when x=33

To find the height of the arch of the "Hatch Shell" 33 feet away from the midpoint of the base, we can use the equation of an ellipse.

The equation of an ellipse with a horizontal major axis and centered at the origin is given by:

x^2/a^2 + y^2/b^2 = 1

Where "a" is the semi-major axis and "b" is the semi-minor axis of the ellipse.

In this case, the length of the base of the arch is the length of the semi-major axis (a). So, a = 110/2 = 55 feet.

The highest part of the arch is the value of the y-coordinate when x = 55/2 = 27.5 feet. We are given that the highest part of the arch is 40 feet above the scene. So, the value of b is equal to the height of the arch plus the highest part of the arch, which is 40 + 27.5 = 67.5 feet.

Now, we can find the equation of the ellipse using the known values of a and b:

x^2/(55/2)^2 + y^2/67.5^2 = 1

To find the height of the arch 33 feet away from the midpoint of the base, we substitute x = 33 into the equation and solve for y.

(33)^2/(55/2)^2 + y^2/67.5^2 = 1

Solving this equation will give us the value of y, which represents the height of the arch at a distance of 33 feet away from the midpoint of the base.

To find the height of the arch of the "Hatch Shell" 33 feet away from the midpoint of the base of the arch, we can use the equation for the shape of a semi-elliptical arch. Here's how you can solve this problem:

Step 1: Draw a diagram: Start by drawing a diagram that represents the arch with a horizontal major axis, with the base length of 110 feet and the highest point 40 feet above the base.

Step 2: Identify the key points: Label the midpoint of the base as the point A. The highest point of the arch can be labeled as point B. The point 33 feet away from the midpoint (point A) on one side of the arch can be labeled as point C.

Step 3: Determine the coordinates: Assign coordinates to the key points. Let the x-coordinate of point A be 0. Therefore, the x-coordinate of point C will be 33 feet.

Step 4: Use the equation for a semi-elliptical arch: The equation for a semi-elliptical arch with a horizontal major axis is y = (b^2 - (b^2/a^2) * x^2)^(0.5), where (a, b) are the semi-major and semi-minor axes of the ellipse.

Step 5: Calculate the semi-major and semi-minor axes: The semi-major axis (a) is half the length of the base, so a = 110/2 = 55 feet. The semi-minor axis (b) is the height of the arch, so b = 40 feet.

Step 6: Plug in the values: Substitute the values of a, b, and x into the equation from step 4. For point C, we have x = 33.

y = (b^2 - (b^2/a^2) * x^2)^(0.5)
y = (40^2 - (40^2/55^2) * 33^2)^(0.5)
y = (1600 - (14400/3025) * 1089)^(0.5)
y = (1600 - (633.06) * 1089)^(0.5)
y = (1600 - 717684.84)^(0.5)
y = (-716084.84)^(0.5)
* Since we cannot take the square root of a negative number, we conclude that there is no arch at that point.

Therefore, the height of the arch at 33 feet away from the midpoint of the base is undefined or non-existent.