The equation of the ellipse with height = 26 cm and width = 16 cm is - [tex]\frac{x^{2} }{169 } +\frac{y^{2} }{64} } =1[/tex] and the horizontal width, in centimeters, of the plaque at a distance of 6 centimeters above the center point is 9.81cm
We have a wooden plaque in the shape of an ellipse with height 26 centimeters and width 16 centimeters.
We have to find equation for the ellipse and use it to find the horizontal width, in centimeters, of the plaque at a distance of 6 centimeters above the center point.
The general form of the equation of ellipse is -
[tex]\frac{x^{2} }{a^{2} } +\frac{y^{2} }{b^{2} } =1[/tex]
Where -
a is the length of semi major axis.
b is the length of semi minor axis.
In the question given to us -
height of ellipse = length of major axis = 2a =26
width of ellipse = length of minor axis = 2b = 16
Therefore -
a = 13 cm
and
b = 8 cm
Substituting the values in the equation of ellipse, we get -
[tex]\frac{x^{2} }{(13)^{2} } +\frac{y^{2} }{(8)^{2} } =1[/tex]
[tex]\frac{x^{2} }{169 } +\frac{y^{2} }{64} } =1[/tex]
Now, at y = 6 cm above the center -
[tex]\frac{x^{2} }{169 } = 1 - \frac{36 }{64} }\\\\\frac{x^{2} }{169 } = 0.57\\x^{2} = 169\times0.57\\x = 9.81 cm[/tex]
Hence, the equation of the ellipse with height = 26 cm and width = 16 cm is - [tex]\frac{x^{2} }{169 } +\frac{y^{2} }{64} } =1[/tex] and the horizontal width, in centimeters, of the plaque at a distance of 6 centimeters above the center point is 9.81cm
To solve more questions on Ellipse, visit the link below -
https://brainly.com/question/2660421
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