Answer:
The value is ![C \approx \$76](https://tex.z-dn.net/?f=C%20%5Capprox%20%5C%2476)
Step-by-step explanation:
From the question we are told that
The area of the rectangular pen is ![A = 24 \ yard^2](https://tex.z-dn.net/?f=A%20%3D%20%2024%20%5C%20yard%5E2)
The cost of material used to make one side is ![z = \$ 6](https://tex.z-dn.net/?f=z%20%3D%20%5C%24%206)
The cost of material used to make the other sides is ![r = \$ 3](https://tex.z-dn.net/?f=r%20%20%3D%20%20%5C%24%203)
Now , the fence to be build around the rectangular pen has four sides, the first opposite sides are equal, let assume each of the to be x yard and the other opposite sides are also equal as well let assume of the to be y yard
So the cost is mathematically represented as
![C = zx + r (x + 2y )](https://tex.z-dn.net/?f=C%20%3D%20%20zx%20%20%2B%20%20r%20%28x%20%2B%202y%20%29)
=> ![C = 6x + 3(x + 2y)](https://tex.z-dn.net/?f=C%20%3D%20%206x%20%20%2B%20%203%28x%20%2B%202y%29)
=> ![C = 9x + 6y](https://tex.z-dn.net/?f=C%20%3D%20%20%209x%20%20%2B%20%206y)
Now the area of the fence is mathematically represented as
![A = x* y = 24](https://tex.z-dn.net/?f=A%20%3D%20%20x%2A%20y%20%3D%20%2024)
=> ![y = \frac{24}{x}](https://tex.z-dn.net/?f=y%20%20%3D%20%20%5Cfrac%7B24%7D%7Bx%7D)
=> ![C = 9x + 6[\frac{24}{x} ]](https://tex.z-dn.net/?f=C%20%3D%20%20%209x%20%20%2B%20%206%5B%5Cfrac%7B24%7D%7Bx%7D%20%5D)
=> ![C = 9x + [\frac{144}{x} ]](https://tex.z-dn.net/?f=C%20%3D%20%20%209x%20%20%2B%20%20%5B%5Cfrac%7B144%7D%7Bx%7D%20%5D)
Now differentiating
![C' = 9 + 144* (-2) x^{-2}](https://tex.z-dn.net/?f=C%27%20%20%3D%20%209%20%2B%20144%2A%20%20%28-2%29%20x%5E%7B-2%7D)
![C' = 9 - 288x^{-2}](https://tex.z-dn.net/?f=C%27%20%20%3D%20%209%20-%20288x%5E%7B-2%7D)
At minimum ![C' = 0](https://tex.z-dn.net/?f=C%27%20%3D%20%200)
So
![9 - 288x^{-2} = 0](https://tex.z-dn.net/?f=9%20-%20288x%5E%7B-2%7D%20%3D%20%200)
![x^{-2} = 0.03125](https://tex.z-dn.net/?f=x%5E%7B-2%7D%20%3D%20%200.03125)
![x = \sqrt{\frac{1}{0.03125} }](https://tex.z-dn.net/?f=x%20%3D%20%20%5Csqrt%7B%5Cfrac%7B1%7D%7B0.03125%7D%20%7D)
![x = 5.66](https://tex.z-dn.net/?f=x%20%3D%20%205.66)
Now substituting for x in the equation above to obtain minimum cost
![C = 9(5.66) + [\frac{144}{5.66} ]](https://tex.z-dn.net/?f=C%20%3D%20%20%209%285.66%29%20%20%2B%20%20%5B%5Cfrac%7B144%7D%7B5.66%7D%20%5D)
![C \approx \$76](https://tex.z-dn.net/?f=C%20%5Capprox%20%5C%2476)