First, apply the formula of area of rectangle to express the dimension of the side made out of steel in terms of x.
`A= l e n g t h * width`
Then, plug-in length=x and A=200.
`200= x* width`
And, divide both sides by x to isolate width.
`200/x = width`
Hence, the sides of the rectangle are x (aluminum) and 200/x (steel).
So, to get the cost function of the pen, multiply each sides by of the rectangle by the unit price of the material.
Hence, the cost function of the pen in terms of x is `C(x) =26x+7200/x ` .
(b) To solve, take the derivative of the cost function.
`C'(x) = (26x+7200/x)' = 26 - 7200x^(-2)`
Then, set C'(x) equal to zero.
To simplify, multiply both sides by x^2.
Then, use the quadratic formula to solve for x.
Since x represents the length of the side made of aluminum, take only the positive value. So,
Now that the value of x is known, determine the length of the side made of steel. To do so, plug-in the value of x to:
Hence, the dimension of the rectangular pen that will minimize the cost is `16.6 xx 12.0` feet.