Question

A rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides. If the total area is 384 square feet, find the dimensions of the study area that will minimize the total length of the fence. How much fencing will be required?

Answer 1

Answer:

Explanation:

Let l be the length of the study area and w be the width of study area.

Area of rectangular part = l x w

A = l x w

384 = l x w

w = 384 / l .... (1)

length of the fence is equal to the perimeter.

P = 2 ( l + w)

$P = 2\left ( l +\frac{384}{l} \right )$   from equation (1)

Differentiate with respect to l

$\frac{dP}{dl} = 2\left ( 1 -\frac{384}{l^{2}} \right )$

for maxima and minima, it is equal to zero

$\left ( 1 -\frac{384}{l^{2}} \right )=0$

l = 19.6 ft

so w = 19.6 ft

So, the required fencing = 2 ( 19.6 + 19.6) = 78.4 ft

Answer 2

The amount of fence that will be required is 78.4 feet

The formula for calculating the perimeter of the rectangular study area is expressed as;

$P=2(l+w)$

The formula for calculating the area of the rectangular study is expressed as:

$A =lw\\w=\frac{A}{l}$

Given the total area is 384 square feet, hence $w=\frac{384}{l}$

Substitute the expression for the width into the perimeter is expressed as:

$P=2(l+\frac{384}{l} )$

If the study area will minimize the total length of the fence, then $\frac{dP}{dl} = 0$

$\frac{dP}{dl}=2(1-\frac{384}{l^2} ) \\2(1-\frac{384}{l^2} ) =0\\\frac{l^2-384}{l^2}=0\\l^2-384=0\\l^2=384\\l=\sqrt{384}\\l = 19.6feet$

To get the amount of fencing that will be required, we will find the perimeter of the rectangular study area

$P=2(19.6+19.6)\\P=2(39.2)\\P=78.4feet$

Hence the amount of fence that will be required is 78.4 feet

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