Answer:
Side opposite the river = 120 ft
Other sides = 240 ft
Explanation:
Let 'R' denote the length of fence opposite to the river and 'L' denote the length of the other two sides.
The cost as a function of R is:
[tex]L*R = 28,800\\L=\frac{28,800}{R}\\ C = 40R+10*2*\frac{28,800}{R} \\C(R) = 40R+576,000R^{-1}[/tex]
The value of R for which the derivate of the cost function is zero is the length that minimizes cost:
[tex]C'(R) =0= 40 -576,000R^{-2}\\R=\sqrt{\frac{576,000}{40}}\\R=120\ ft\\[/tex]
If R is 120 ft, then the value of L is:
[tex]L = \frac{28,800}{120}\\L=240\ ft[/tex]
The dimensions that will minimize costs are:
Side opposite the river = 120 ft
Other sides = 240 ft
