A rectanglar plot of land is to be fenced in using two kinds of fencing. Two opposite sides will use heavy-duty fencing selling for $6 a foo, while the remaining two sides will use the standard fencing selling at $4 a foot. What are the dimensions of the rectanglar plot of greatest area that can be fenced at a cost of $12000?

250 x 1125ft

Step-by-step explanation:

Let x and y be the sides of the rectangle.

Then, set-up the equation for the total cost of the fence.

To do so, given that $6 per foot fencing for the two sides "x" . And, $4 per foot fencing for the two sides "y".

So the equation is

12000 = 6(2x) + 4 (2y)

12000 = 12x + 8y

To simply, divide both sides by its GCF which is 2

12000/2 = (12x + 8y)/2

6000 = 6x + 4y

Then solve for y

6000-6x = 4y

=> (6000-6x)/4 = y

=> 1500 - (6/4)x = y

Since we have to maximize the area of the rectangle, set up the equation of the area.

Area = xy

Note that y = 1500 - (6/4)x

Therefore, A = x (1500 -6/4x)

A=1500x - 6/4x^2

Then, take the derivative of A.

A' =1500 - 6x

Set A' equal to zero and solve for x.

0 = 1500-6x

6x = 1500

x = 250

Substitute the value of x to y = 1500 - 6/4x .

y = 1500 - 6/4 * 250 = 1500 - 375 = 1125

Hence, the dimensions of the rectangular plot that would maximize its area, given the total cost of fencing, is 250 x 1125 ft.

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1

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step-by-step explanation: