022 Exam 2 Sample B, Problem 10

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Use calculus to set up and solve the word problem: A fence is to be built to enclose a rectangular region of 480 square feet. The fencing material along three sides cost $2 per foot. The fencing material along the 4th side costs $6 per foot. Find the most economical dimensions of the region (that is, minimize the cost).

As with all geometric word problems, it helps to start with a picture. Using the variables and as shown in the image, we need to remember the equations of a rectangle for area:
However, we need to construct a new function to describe cost:
Since we want to minimize cost, we will have to rewrite it as a function of a single variable, and then find when the first derivative is zero. From this, we will find the dimensions which provide the minimum cost.


Step 1:  
Express one variable in terms of the other: Since we know that the area is 480 square feet and , we can solve for in terms of . Since , we find that .
Step 2:  
Find an expression for cost in terms of one variable: Now, we can use the substitution from step 1 to find
Step 3:  
Find the derivative and its roots: We can apply the power rule term-by-term to find
This derivative is zero precisely when , which occurs when , and these are the values that will minimize cost. Also, don't forget the units - feet!
Final Answer:  
The cost is minimized when the dimensions are feet by feet. Note that the side with the most expensive fencing is the shorter one.

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