Answer: x^2 / (8 - 2sqrt(190))^2 + y^2 / 1225 = 1
Explanation:
To find the equation of the semiellipse, we need to determine the horizontal and vertical radii of the ellipse. We know that the center of the ellipse is at the origin, so we can use the standard form of the equation for a horizontal ellipse:
x^2/a^2 + y^2/b^2 = 1
where a is the horizontal radius and b is the vertical radius.
From the problem statement, we know that the height of the tunnel at the center is 35 ft, so the vertical radius is b = 35 ft. We also know that the vertical clearance must be 21 ft at a point 8 ft from the center, which means that the distance from the center to this point is 8 ft plus the horizontal radius. Using the Pythagorean theorem, we can write:
(8 + a)^2 + 21^2 = 35^2
Expanding and simplifying, we get:
64 + 16a + a^2 + 441 = 1225
a^2 + 16a - 720 = 0
We can solve this quadratic equation using the quadratic formula:
a = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 16, and c = -720. Plugging in these values, we get:
a = (-16 ± sqrt(16^2 - 4(1)(-720))) / 2(1)
a = (-16 ± sqrt(16^2 + 4(1)(720))) / 2
a = (-16 ± sqrt(3040)) / 2
We can simplify this expression by factoring out 4 from the radical:
a = (-16 ± 4sqrt(190)) / 2
a = -8 ± 2sqrt(190)
Since a is the horizontal radius of the ellipse, we take the positive value:
a = -8 + 2sqrt(190)
Therefore, the equation of the semiellipse is:
x^2 / (8 - 2sqrt(190))^2 + y^2 / 35^2 = 1
Simplifying the expression under the square root:
x^2 / (8 - 2sqrt(190))^2 + y^2 / 1225 = 1
This is the equation of the semiellipse.