A boy wants to build a rectangular pen that is made out of s

A boy wants to build a rectangular pen that is made out of steel on two(opposite) sides and aluminum on the other two sides for his science project. The steel fencing costs $18 per linear foot and the aluminum fencing is $13 per linear foot. The pen needs to contain 200 square feet of space.

a. Let x be the length of one of the aluminum sides. Find a function for the cost of the pen in terms of x

b. Find the dimensions of the pen that will minimize the cost (to the nearest tenth if needed).

Solution

(a)
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.

[C= 13x+13x+18*200/x+18*200/x]

[C=26x+7200/x]

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)]

[C\'(x)=26-7200/x^2]

Then, set C\'(x) equal to zero.

[0=26-7200/x^2]

To simplify, multiply both sides by x^2.

[0=26x^2-7200]

Then, use the quadratic formula to solve for x.

[x=(-b+-sqrt(b^2-4ac))/(2a)=(-0+-sqrt(0^2-4(26)(-7200)))/(2*26)]

[x=(+-sqrt748800)/52]

[x=+-16.64]

Since x represents the length of the side made of aluminum, take only the positive value. So,

[x=16.6]

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:

[200/x=200/16.6 =12.0]

Hence, the dimension of the rectangular pen that will minimize the cost is [16.6 xx 12.0] feet.