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Dahasolnce [82]
3 years ago
11

You have 600 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river

, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Mathematics
1 answer:
inn [45]3 years ago
7 0

Answer:

The length and width of plot is L=300\:ft, W=150\:ft

Largest area of the plot is A=45000 ft^{2}

Step-by-step explanation:

Assume width as x and length as y. Given that length of fencing is 600 feet and fencing is enclosed on 3 sides. So perimeter is given as,

Perimeter = width + length + width

Substituting the value,  

600=x+y+x

600= 2x+y  ….1

Now area of fence of rectangular box is given as follows,  

A=xy  ….2

Solving equation 1 for y, subtracting 2x from both sides,  

600-2x= y

Substituting the value in equation 2,  

A=x\left (600-2x  \right )

Simplifying

A=600x-2x^{2}

Rewriting,  

A=-2x^{2}+600x

Above equation looks like quadratic equation f\left ( x \right )=ax^{2}+bx+c whose graph looks like parabola.  

Comparing equation f(x) and A values of a, b and c are, a=-2,b=600 and c=0.  

Now maximum of f\left ( x \right ) occurs at vertex.  

The x coordinate of the vertex is given as -\dfrac{b}{2a}

Substituting the values,  

x=-\dfrac{600}{2\left (-2  \right )}

Simplifying,  

x=\dfrac{600}{4}

x=150

So width of plot is 150 feet.  

Now to calculate value of length by using equation 1,

600= 2x+y

Substituting the values,

600= 2\left (150  \right )+y

600= 300+y

Subtracting 300 from both sides,

300= y

So length of plot is 300 ft.

The y coordinate of the vertex is given as y=f\left ( -\dfrac{b}{2a} \right ) which also means, y=f\left ( 150 \right )

\therefore A=-2\left (150  \right )^{2}+600\left (150  \right )

Simplifying,  

\therefore A=-2\left (22500  \right )^{2}+600\left (150  \right )

\therefore A=-45000+90000

\therefore A=45000

So, area of the plot will be A=45000 ft^{2}

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45.96               38.76               2112.3216     1502.3376          1781.4096

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52.17                44.34              2721.7089      1966.0356        2313.2178

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