a. The equation of the ellipse is x²/900 + y²/400 = 1
b. the foci of the ellipse are at (±10√5,0)
c. Find the graph in the attachment
How to find the equation of the ellipse?
Since the ellipse is made from a block of wood and centered at the origin, and x- axis major axis.
So, the equation of an ellipse with center at origin and x - axis major axis is
x²/a² + y²/b² = 1 where
- 2a = length of major axis
- 2b = length of minor axis and
- a > b
Since the rectangular piece of wood has a length of 60 inches and a width of 40 inches.
The length of the rectangle equals the length of the major axis
So, 2a = 60
a = 60/2
= 30
Also, the width of the rectangle equals the length of the minor axis
So, 2b = 40
b = 40/2
= 20
a. The equation of the ellipse
So, substituting a and b into the equation of the ellipse, we have
x²/a² + y²/b² = 1
x²/30² + y²/20² = 1
x²/900 + y²/400 = 1
So, the equation of the ellipse is x²/900 + y²/400 = 1
b. The foci of the ellipse
The foci of the ellipse are at (±c,0) where c² = a² - b²
= 30² - 20²
= 900 - 400
= 500
c = ±√500
c = ±10√5
So, the foci of the ellipse are at (±10√5,0)
c. Find the graph in the attachment
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