Question

The arch given below is in form of a semi-ellipse. It is 8 m wide and its height at the centre is 2 m. Find the height of the arch at a distance of 1.5 m from one end.

Solution

Width of the elliptical arch = 8 m $rightwards double arrow$ AA' = 8m $rightwards double arrow$ 2a = 8m $rightwards double arrow$ a = 4
Height at the centre = 2m $rightwards double arrow$ OB = 2m $rightwards double arrow$ b = 2
∴ the equation of ellipse is $straight x squared over 16 plus straight y squared over 4 equals 1$ ...(i)
$AP space equals space 1.5 space straight m space rightwards double arrow space OP space equals space OA minus AP space equals space 4 minus 1.5 equals 2.5 space straight m$
Let the height at P be h.
∴ Coordinates of L are (2.5, h)
It lies on ellipse (i) $rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
$rightwards double arrow$ $straight h squared equals fraction numerator 16 minus 6.25 over denominator 4 end fraction equals fraction numerator 9.75 over denominator 4 end fraction equals 2.44 space left parenthesis Approx right parenthesis$
Hence, $straight h equals square root of 2.44 end root space equals space 1.56 space straight m space left parenthesis Approx. right parenthesis$

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Binomial Theorem

The arch given below is in form of a semi-ellipse. It is 8 m wide and its height at the centre is 2 m. Find the height of the arch at a distance of 1.5 m from one end.

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