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A rancher has 200 feet of fencing to enclose two adjacent rectangular corrals (see figure).

(a) Write the area $ A $ of the corrals as a function of $ x $.(b) Create a table showing possible values of $ x $ and the corresponding areas of the corral. Use the table to estimate the dimensions that will produce the maximum enclosed area(c) Use a graphing utility to graph the area function. Use the graph to approximate the dimensions that will produce the maximum enclosed area.(d) Write the area function in standard form to find analytically the dimensions that will produce the maximum area.(e) Compare your results from parts (b), (c), and (d).

a) $\frac{8 x(50-x)}{3}$b) $25, \frac{100}{3}$c) $25, \frac{100}{3}$d) $x=25 \mathrm{ft}, y=\frac{100}{3} \mathrm{ft}$e) same

Algebra

Chapter 2

Polynomial and Rational Functions

Section 1

Quadratic Functions and Models

Quadratic Functions

Complex Numbers

Polynomials

Rational Functions

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We are discussing numerical graphical and and analytical analyses. So we have a rancher. Is 200 ft of fencing too close to adjacent rectangular corrals as shown in the figure from your textbook? So part A. We need to write the area of the carouse is a function of X, so in order to write it as a function of X, let's first look at the perimeter so the perimeter is equal. Thio four x plus three y and we're told that the perimeter is equal to 200. So now solving for why that why is equal to 200 minus four x over three. So now that we have, why is a function of X? We will use that to find the area so the area is equal to two x times y So this is two x Times 200 minus four x over three. So this tells us that the area is equal to eight x times 50 96 over three. Here we have created a table that shows the possible values of X and the corresponding areas of the corral. So to estimate the dimensions that will produce the maximum enclosed area, we can see that when X equals 20 in the area is 1600 when X equals 25 then the area is 16, 66.67 and then at 38 goes back down. So from the table weaken, see that X equals 25 will give us a maximum. So now let's look at the graph of the function to the right for part C, so at the maximum, the maximum is at X equals 25. So comparing our results from heart being part C finding a table or looking at a graph, they both will give us an estimate for the maximum enclosed area.

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