Arectangular garden of area 832 square feet is to be surrounded on three sides by a brick wall costing $8 per foot and on one side by a fence costing $5 per foot. find the dimensions of the garden such that the cost of the materials is minimized.

The dimensions of the garden such that the cost of the materials is minimized are 26 feet, 32 feet.

Let the length and the breadth of the rectangular garden be x and y respectively.

Area of a rectangle =

832 square feet=

⇒

Assume the side y is a fence, and the rest brick.

Total cost,

⇒

Differentiate the cost with respect to y and equate the derivative C'(y) to zero for maximum or minimum value of y.

Therefore the length of the rectangular garden is

And the second derivative is

So the cost is minimum at y=32 feet

Therefore the dimensions of the garden are x=26 feet, y=32feet.

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step-by-step explanation:

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