Question 1132214
<br>
With a fence down the middle to keep the two dogs separate, the fencing must cover the length of the pen twice and the width three times:<br>
{{{2L+3W = 180}}}  -->  {{{L = (180-3W)/2 = 90-(3/2)W}}}<br>
The total area of the pen is length times width:<br>
{{{A = LW = W(90-(3/2)W) = 90W-(3/2)W^2}}}<br>
If you know calculus, find where the derivative of the area function is zero:<br>
{{{dA/dW = 90-3W}}}
{{{90-3W = 0}}}
{{{W = 30}}}<br>
The maximum area is when W = 30, which makes L = 90-(3/2)(30) = 45.<br>
If you don't know calculus, you can find the maximum area by finding the vertex of the parabolic area function.<br>
{{{A = 90W-(3/2)W^2}}}
{{{W = -b/(2a) = -90/-3 = 30}}}<br>
Then, as before, W = 30 leads to L = 45.<br>
So the dimensions for the greatest total area are 30 feet by 45 feet; the total area is 30*45 = 1350 square feet.