A rectangular doorway is 38 meters wide and 60 meters high. The doorway is capped with a semi-ellipse that is 13 meters high at its center. Find the height h of the doorway and window 5 meters from the center.
From the ellipse equation, I can see that a should be 19 and b should be 13. From the description in the word problem, it sounds like the top half of an ellipse, so it sounds like a semi ellipse equation should be written x = -square root of the ellipse equation. I'm at a loss to find the equation or to solve the question. Please help! Thank you.
So if you place the ellipse with centre at (0,0), we have
a=19 and b = 13 , and the equation would be
x^2/19^2 + y^2/13^2 = 1
or
x^2/361 + Y62/169 = 1
Now if x = 5
25/361 + y^2/169 = 1
y^2/169 = 336/361
y^2 = 157.296...
y = 12.54
adding on the 60 m it would be 72.54 m high.
Wow, curious where we could find that structure.
To find the height h of the doorway and window 5 meters from the center, we can use the equation of a semi-ellipse.
In this case, the semi-ellipse has a height of 13 meters at its center, which means the semi-major axis a is 19 meters (half of the width of the doorway, which is 38 meters).
The equation of an ellipse centered at the origin is:
(x^2 / a^2) + (y^2 / b^2) = 1,
where a is the semi-major axis length and b is the semi-minor axis length.
Since we are interested in the height h of the doorway and window, which corresponds to the y-coordinate, we can rearrange the equation as:
y = b * sqrt(1 - (x^2 / a^2)).
Substituting the values a = 19 and b = 13 into the equation, we have:
y = 13 * sqrt(1 - (x^2 / 19^2)).
To find the height h, we need to find the value of y when x = 5. Plugging x = 5 into the equation, we get:
y = 13 * sqrt(1 - (5^2 / 19^2)).
Calculating this expression will give us the height h of the doorway and window 5 meters from the center.