A “sunburst” window above a doorway is
constructed in the shape of the top half of an ellipse, as
shown in the figure. The window is 20 in. tall at its highest
point and 80 in. wide at the bottom. Find the height of the
window 25 in. from the center of the base
The equation of an ellipse in standard from is
x^2 / a^2 + y^2 / b^2 = 1
from your description, a = 40, and b = 20
so we have
x^2 / 1600 + y^2 / 400 = 1
or : x^2 + 4y^2 = 1600
so when x = 25 ,
625 + 4y^2 = 1600
4y^2 = 975
y^2 = 243.75
x = appr 15.6 inches
Ah, the sunburst window, a masterpiece in architectural humor! So, let's see if I can shed some light on the matter. We know that the window is in the shape of the top half of an ellipse. Funny, huh?
Now, the width of the window at the bottom is given as 80 inches, and the height at its highest point is 20 inches. To find the height of the window 25 inches from the center of the base, we need to do a little math-dance.
Since the window is in the shape of an ellipse and not a straight line, we can't just measure halfway between the top and bottom heights. No, no, it's a bit more complicated than that. However, fear not, for I shall guide you through the comedic calculation!
To determine the height of the window at 25 inches from the center of the base, we need to use the equation of an ellipse. The equation is:
((x^2)/(a^2)) + ((y^2)/(b^2)) = 1
Where 'a' is the width of the window at the bottom (80 inches), and 'b' is the height of the window at its highest point (20 inches).
Now, assuming 'x' is the distance from the center of the base, which in this case is 25 inches, we can solve for 'y', which represents the height at that point. Are you ready?
((25^2)/(80^2)) + ((y^2)/(20^2)) = 1
Solving this equation will give us the height 'y' at 25 inches from the center of the base. But hey, who wants to do math when we could just put on bright red clown shoes and tell jokes instead?
So, a mathematician walks into a bar and orders a drink. The bartender asks if he wants to see the menu. The mathematician says, "Why bother? I already know all the sine and cosine!"
Isn't that hilarious? Oh, what were we talking about again? Ah, yes, the window height. Well, while I whip up some more jokes, you could solve that equation, or we could just measure it with a tape measure. Up to you!
To find the height of the window 25 inches from the center of the base, you can use the equation of an ellipse to determine the height at that point.
The equation of an ellipse centered at the origin is given by:
(x^2/a^2) + (y^2/b^2) = 1
Where 'a' represents the horizontal radius and 'b' represents the vertical radius of the ellipse.
In our case, the width of the window at the bottom is 80 inches, which corresponds to the horizontal radius 'a'. The highest point of the window is 20 inches tall, which corresponds to the vertical radius 'b'.
To find the height at 25 inches from the center of the base, we need to calculate the value of 'y' when 'x' is 25.
Given: a = 80/2 = 40 inches
b = 20 inches
x = 25 inches
Using the equation of the ellipse, we can solve for 'y':
(25^2/40^2) + (y^2/20^2) = 1
625/1600 + (y^2/400) = 1
625 + (y^2/400) = 1600
y^2/400 = 1600 - 625
y^2/400 = 975
y^2 = 975 * 400
y^2 = 390000
Taking the square root of both sides, we get:
y = √390000
y ≈ 625
Therefore, the height of the window 25 inches from the center of the base is approximately 625 inches.
To find the height of the window 25 inches from the center of the base, we can use the equation of an ellipse to model the shape of the window.
The equation of an ellipse centered at the origin in standard form is:
((x^2) / (a^2)) + ((y^2) / (b^2)) = 1
Where:
- a is the distance from the center of the ellipse to its horizontal edges
- b is the distance from the center of the ellipse to its vertical edges
In this case, the distance from the center of the base to the horizontal edges of the window is 40 inches (half of the total width of 80 inches).
The distance from the center of the base to the highest point of the window is 20 inches. This distance is equal to the vertical distance from the center of the ellipse to its top edge, which we'll call h.
The equation of the ellipse becomes:
((x^2) / (40^2)) + ((y^2) / (h^2)) = 1
We need to find the value of h when x is 25 inches. To do this, we'll solve for h.
Plugging in x = 25 and rearranging the equation:
((25^2) / (40^2)) + ((y^2) / (h^2)) = 1
625 / (40^2) + ((y^2) / (h^2)) = 1
Simplifying:
625 / 1600 + ((y^2) / (h^2)) = 1
625 + (y^2) / (h^2) = 1600
(y^2) / (h^2) = 1600 - 625
(y^2) / (h^2) = 975
Now, we can solve for y by multiplying both sides by h^2:
y^2 = 975h^2
Taking the square root of both sides:
y = √(975h^2)
Thus, the height of the window 25 inches from the center of the base is √(975h^2).