Minimize Fencing for 4 Pens of Given Area
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A pet store is designing 4 pens to be in a row as shown.Each pen has dimensions a by b and should have an area of 40 sq. ft.
Find the dimensions which will minimize the amount of fencing, L, required.
How should we proceed?
We can try to express L in terms of one variable. Then we can determine the value of that variable which minimizes L in our usual way by differentiating, setting that result to zero, and solving for the value which minimizes.
First, let's express L in terms of both a and b, including all four pens.
We need five pieces of fencing that are a long and eight pieces that are b long.
State that mathematically.
Equation 1
Because we have two unknowns, we need another equation relating them. We can get it by expressing the required area of a pen in terms of a and b. Do that.
Equation 2
Both a and b are involved in L in equation 1. How can we get an equation in terms of just one of them?
We can use equation 2 to express one variable in terms of the other one.
Solve equation 2 for a.
Equation 3
Substitute in equation 1.
We get
Equation 4
Differentiate with respect to the width, b
Equation 5
Solve for the value of b which minimizes the length of fencing.
How do we interpret this result?
Both values are extrema of equation 4. Since we want a physical dimension, we discard the –5 value.
Hence
minimizes the length of fencing required.
Are we done?
No.
What else do we need to determine?
We need the value of a that corresponds to the width we just found. How can we find that?
We can substitute the width into equation 3. Do that.
We get
So how much fencing is needed? How can we get that?
We can use equation 1. Do that.
ft
of fencing is required to create the four pens which each are 5 ft. by 8 ft. in size.
How can we check that this is a minimum and not a maximum?
We can find the second derivative of L from equation 5, and evaluate it at b = 5. Then we can check the algebraic sign.
Do that.
What is the algebraic sign of the second derivative?
Since we have a positive value of b, the sign is positive.
Is that consistent with a minimum?
Yes.
Draw a graph of equation 4 to also confirm that this is a minimum. Then check with the graph below.
Graph of the length of fencing versus the width of a pen
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