Question 1185272
.
A school is building a new track for cycling teams. The track is to be elliptical,
with the ratio between the lengths of the major and minor axes to be 4:3. If the available
land is 200 yd long and 100 yd wide, what are the maximum lengths of the axes? Sketch
a graph of the track and indicate the location of the foci.
~~~~~~~~~~~~~~~


<pre>
Let  "a"  be the major semi-axis of the ellipse;  "b"  be the minor semi-axis.


From the condition, we have 


   a = 4x;  b = 3x,   where x is the common factor

   4x <= 100 yards    (one half of 200 yards);    (1)

   3x <=  50 yards    (one half of 100 yards).    (2)


From restrictions (1) and (2)

   
    x <= 25  yards;  

    x <= 16 {{{2/3}}} yards.


According to the meaning of the variables, we must take the minimum value of (3) and (4) for  x;  so, we take

    
    x = 16 {{{2/3}}}  yards.


Thus  a = 4x = 64 {{{8/3}}} = 66 {{{2/3}}} yards.

      b = 3x = 50 yards.


The distance from the center of the ellipse to the foci is


     c = {{{sqrt(a^2-b^2)}}} = {{{sqrt(66.666^2 - 50^2)}}} = 44.095 yards.
</pre>

Solved.