Find dimensions of the corral with maximum area.?

Question

Suppose that 600 ft of fencing are used to enclose a corral in the shape of a rectangle with a semicircle whose diameter is a side of the rectangle. Find the dimensions of the corral with maximum and area.

Answer 1

i am not able to clearly picture your shape here ... do you have a half circle shaped corral? half circle sitting on top of a rectangle?

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solution ideas -- shape with maximum area for minimum perimter is a circle.

square (90 degree) corner shape with maximum area for minimum perimeter is the one that most closely approximates a circle -- ie; a square

it follows that if your shape is a half circle on top of a rectangle, the winning shape is a half circle mounted on top of a square [square is a special case of a rectangle in which all the sides are of equal length].

the total perimeter of that half circle on a square shape would be 3x [three sides of the square] plus 1/2 * pi * x. [one half the perimeter of a circle -- whole circle is 2 pi radius but your half circle's flat side is 2r (a diameter)], total of approximately 4.5708 x.

then, if total perimeter is 600, x = 131.27 feet.

the total area is then the area of the square [131.27 squared = 17213.56 sq. ft] plus 1/2 the area of the circle [1/2 times pi times diameter squared divided by 4 -- since area of circle is pi times radius squared and radius = 1/2 diameter -- or about 6766.81 sq ft]. A total of 23980.37 sq. ft.

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now -- here's the real trick. When your teacher calls on you to explain your calculations, will you be able to do it??

Answer 2

enable x be the width of the corral (and diameter of the semicircle) and y the top of the corral. the area enclosed is A = xy + ¼?x² the fringe is P = x+2y + ½? x = six hundred, which we are in a position to view as a constraint equation between x and y. -- you may sparkling up the problem using a Lagrange undetermined multiplier, or replace out y from the constraint equation. i'm going to apply the Lagrangian multiplier. finding the turning element of f(x,y) problem to the constraint g(x,y)=consistent is comparable to finding the minimum/optimal of F(x,y) = f(x,y) + ? g(x,y) problem to the constraint equation, the place ? is a few undetermined consistent. on your occasion F = xy + ¼?x² + ?[ (a million+½?) x + 2y ] ?F/?x = y + ½?x + ? (a million+½?) [A] ?F/?y = x + 2? [B] the two equations would desire to be 0 for a minimum/optimal [B] tells us that ? = -x/2 and substituting in [A] supplies y + ½?x - ½x (a million+½?) = 0 => y = ½x - ¼? x [C] So 2y = x - ½?x This would desire to fulfill the constraint equation. Substituting for 2y in P supplies 2 x = six hundred => x = 3 hundred ft and from [C] y = one hundred fifty - 75? ft --