A fence must be built to enclose a rectangular area of $45,000 \mathrm{ft}^{2}$. Fencing material costs $\$ 4$ per foot for the two sides facing north and south and $\$ 8$ per foot for the other two sides. Find the cost of the least expensive fence.
The cost of the least expensive fence is $\$ \square$. (Simplify your answer.)

Step 1 ：Let the length of the enclosure be \(l\) and the width be \(w\). We have the equation \(lw=45000\). We want to minimize the cost of this rectangular fence, which is given by \(4l+8w\). From our equation, we know that \(l=45000/w\). Substituting this into our expression for cost, we have \[(4(45000/w)+8w)=180000/w+8w\]

Step 2 ：We will now find the minimum value of this expression. Taking the derivative of the cost function with respect to \(w\), we get \[-180000/w^2+8\]. Setting this equal to zero and solving for \(w\), we get \[w^2=22500\] or \[w=150\].

Step 3 ：Substituting \(w=150\) back into the equation \(lw=45000\), we get \[l=45000/150=300\].

Step 4 ：Hence, the cost of the least expensive fence is \(4l+8w=4*300+8*150=\boxed{2400}\) dollars.