Answer:
The least cost of fencing for the rancher is $1200
Step-by-step explanation:
Let <em>x</em> be the width and <em>y </em>the length of the rectangular field.
Let <em>C </em>the total cost of the rectangular field.
The side made of heavy duty material of length of <em>x </em>costs 16 dollars a yard. The three sides not made of heavy duty material cost $4 per yard, their side lengths are <em>x, y, y</em>. Thus
We know that the total area of rectangular field should be 2250 square yards,
We can say that
Substituting into the total cost of the rectangular field, we get
We have to figure out where the function is increasing and decreasing. Differentiating,
Next, we find the critical points of the derivative
Because the length is always positive the only point we take is . We thus test the intervals
and
we see that total cost function is decreasing on and increasing on
. Therefore, the minimum is attained at
, so the minimal cost is
The least cost of fencing for the rancher is $1200
Here’s the diagram:
