You need to construct fence around an area of 3136 ft2. What are the dimensions of the rectangular pen that minimize the amount of material needed

The dimensions of the rectangle are needed.

The minimum dimensions are length 56 feet and width 56 feet.

Let be the length of rectangle

be the width of rectangle

The area of the rectangle is

Perimeter is given by

Differentiating with respect to

Equating with zero

Double derivative of perimeter at

Since, it is greater than zero the perimeter will be minimum at

Finding

The minimum dimensions are length 56 feet and width 56 feet.

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Step-by-step explanation:

The the area of the rectangular pen be A = LW

L is the length

W is the width

Given

A = 3136ft²

3136 = LW1

Given the perimeter P = 2L+2W 2

From 1; L = 3136/W

Substitute into 2

P = 2(3136/W)+2W

P = 6272/W + 2W

In order to minimize the amount of material needed, then dP/dW = 0

dP/dW = -6272/W² + 2

0 = -6272/W² + 2

6272/W² = 2

cross multiply

6272 = 2W²

W² = 3136

W = √3136

W = 56ft

Since A = LW

3136 = 56L

L = 3136/56

L = 56ft

Hence the dimensions of the rectangular pen that minimize the amount of material needed is 56ft by 56ft

center (1,3) with radius 6

step-by-step explanation:

we must complete the square to find the center and radius of the circle.

first make sure the x and y squared terms have 1 as their coefficients. we also make sure x and y terms together.

we now create space between the x and y terms with parenthesis.

we complete the square by taking the middle terms -2x and the -6y - divide each and square them.

we add the squares to both sides.

simplify.

and write the quadratics in factored form.

the center is (h,k) or (1,3). the radius is the square root of 36 which is 6.