An arch in the form of a semi ellipse has a span of 10 meters and a central height of 4 m. Find the heights of the arch at a point of 3 meters from the semi minor axis.
3.2 m is my final answer.
x^2/25 +y^2/16=1
y=3 given
then x=3.3m
Well, I must say this question really has me arching my eyebrows! But fear not, for I shall attempt to solve this riddle for you.
Now, with a semi-ellipse arch and a span of 10 meters, we can determine the semi-major axis (a) to be half of the span, which is 5 meters.
Next, we have the central height (h) of the arch, which is 4 meters.
To find the heights of the arch at a point 3 meters from the semi-minor axis, we need to calculate the semi-minor axis (b).
Using the Pythagorean theorem, we can find b using the equation: b^2 = a^2 - h^2.
Substituting the known values, we get: b^2 = 5^2 - 4^2 = 25 - 16 = 9.
Taking the square root of both sides, we find that b = 3 meters.
Now, to find the height at a point 3 meters from the semi-minor axis, we can use the equation for the height of an ellipse at a given distance from the center:
Height = sqrt(b^2 - (b^2/a^2) * x^2),
where x is the distance from the center.
Plugging in the values, we get: Height = sqrt(3^2 - (3^2/5^2) * 3^2) = sqrt(9 - (9/25) * 9) = sqrt(9 - (9/25) * 9) = sqrt(9 - 9/25 * 9) = sqrt(9 - 3.24) = sqrt(5.76) = 2.4.
So, the height of the arch at a point 3 meters from the semi-minor axis is approximately 2.4 meters.
I hope I didn't arch too much confusion upon you with my explanation!
To find the heights of the arch at a point 3 meters from the semi-minor axis, we can start by drawing a diagram to visualize the problem.
Let's consider the semi-ellipse with a span of 10 meters and a central height of 4 meters. The semi-minor axis is half of the span, which is 10/2 = 5 meters.
Next, we can find the equation of the semi-ellipse. The equation of a semi-ellipse centered at the origin can be written as:
(x^2/a^2) + (y^2/b^2) = 1
Where 'a' is the semi-major axis and 'b' is the semi-minor axis. In our case, a = 5 (half of the span) and b = 4 (given central height).
So, the equation of the semi-ellipse is:
(x^2/25) + (y^2/16) = 1
Now, we need to find the heights of the arch at a point 3 meters from the semi-minor axis. To do that, we need to substitute the x-coordinate (3) into the equation and solve for y.
(3^2/25) + (y^2/16) = 1
9/25 + (y^2/16) = 1
Multiplying both sides by 16 to eliminate the fraction:
(16 * 9)/25 + y^2 = 16
144/25 + y^2 = 16
y^2 = 16 - 144/25
y^2 = (400 - 144) / 25
y^2 = 256 / 25
Now, taking the square root of both sides:
y = ± √(256 / 25)
y = ± 16/5
So, the heights of the arch at a point 3 meters from the semi-minor axis are 16/5 and -16/5 meters.
Clearly you have an ellipse with semi-axes
a = 5
b = 4
So, c=3
The latus rectum is the line perpendicular to the semi-major axis, through a focus.
The semi-latus rectum is b^2/a = 16/5