logo

Ask Questions, Get Answers

 
X
 Search
Want to ask us a question? Click here
Browse Questions
Ad
0 votes

The arch of a bridge is in the shape of a semi-ellipse having a horizontal span of $40ft$ and $16ft$ high at the centre. How high is the arch $9ft$ from the right or left of the centre .

Can you answer this question?
 
 

1 Answer

0 votes
Toolbox:
  • Standard forms of equation of the ellipse with major axes 2a, minor axis 2b $(a >b)$ eccentricity e and $b^2=a^2(1-e^2)$ or $e^2=1-\large\frac{b^2}{a^2}$
  • $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$$=1$
  • http://clay6.com/mpaimg/21_2_Toolbox.png
  • Foci$(\pm ae,o),$ center $(0,0)$,vertices $(\pm a,0)$
  • End points of Latus Rectum $\; (ae,\pm \large\frac{b^2}{a})$ and $(-ae,\pm \large\frac{b^2}{a})$
  • Directrices $x=\pm \large\frac{a}{e}$.
  • The major axis is $y=0$ (x- axis) and the minor axes is $x=0$ (y- axis)
  • $\large\frac{x^2}{b^2}+\frac{y^2}{a^2}$$=1$
  • http://clay6.com/mpaimg/21_2_Toolbox1.png
  • Foci$(0,\pm ae),$ center $(0,0)$,vertices $(0,\pm a)$
  • End points of Latus Rectum $(\pm \large\frac{b^2}{a}$$,ae)$ and $(\pm \large\frac{b^2}{a}$$,-ae)$
  • Directrices $y=\pm \large\frac{a}{e}$.
  • The major axis is $x=0$ (y- axis) and the minor axes is $y=0$ (x- axis)
Step 1:
Let $A'A$ be the span of the area.$C$ the centre of the ellipse(the origin) and $B$ the highest point.
$CA=CA'=20=a$ the length of the semi major axis.
$CB=16=b$ is the length of the semi minor axis.
Step 2:
The equation of the ellipse is $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}=$$1$
Substitute the values of $'a'$ and $'b'$ we get,
$\large\frac{x^2}{400}+\frac{y^2}{256}$$=1$
Step 3:
Let $PM$ be the position $9$ feet to the right of $C$.
$P$ has coordinates $(9,y)$ substituting we get
$\large\frac{81}{400}+\frac{y^2}{256}$$=1$
$\large\frac{y^2}{256}=1-\large\frac{81}{400}=\large\frac{319}{400}$
$y^2=\large\frac{256\times 319}{400}$
The height PM=$\large\frac{\sqrt{256\times 319}}{400}$
$\Rightarrow \large\frac{16}{20}$$\sqrt{319}$
$\Rightarrow \large\frac{4}{5}$$\sqrt{319}$ft
answered Jun 18, 2013 by sreemathi.v
edited Jun 18, 2013 by sreemathi.v
 

Related questions

Ask Question
student study plans
x
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App
...