Question

Assume you have 12 yards of fencing material available What is the largest area, measured in square yards, of a rectangular enclosure that has a fenced center divider (so that the final enclosure looks like a domino from above)

Answer 1

Answer:

The maximum area is when the largest side of the rectangle is 3 yards and the shortest side of the rectangle is 2 yards.

Step-by-step explanation:

If the final enclosure will look froma above like a domino (rectangular with a center divider).

If we name a and b the sides of the rectangle, we have the area as:

$S=a\cdot b$

If we divide the center of the rectangle with a size b fence (one of the sides of the rectangle), the perimeter used is:

$P=2a+2b+b=2a+3b=12\\\\a=\frac{12-3b}{2}=6-1.5b$

To maximize the area, we derive and equal to zero (after replacement of a by the last equation):

$S=(6-1.5b)\cdot b=-1.5b^2+6b\\\\dS/db=-1.5*2*b+6=0\\\\-3b+6=0\\\\b=6/3=2\\\\a=6-1.5b=6-1.5*2=6-3=3$

The maximum area is when the largest side of the rectangle is 3 yards and the shortest side of the rectangle is 2 yards.

Answer 2

The area of the rectangular enclosure is the product of its side lengths

The largest area of the rectangular enclosure is 6 square yards

In the final enclosure, the area is divided into two equal parts.

Where the perimeter is:

$\mathbf{P = 2a + 3b}$

And the area is

$\mathbf{A = ab}$

The perimeter is given as 12.

So, we have:

$\mathbf{2a + 3b =12}$

Make a the subject

$\mathbf{a =\frac{12 - 3b}2}$

Substitute $\mathbf{a =\frac{12 - 3b}2}$ in $\mathbf{A = ab}$

$\mathbf{A = \frac{12 - 3b}2 \times b}$

$\mathbf{A = \frac{12b - 3b^2}2}$

Multiply through by 2

$\mathbf{2A = 12b - 3b^2}$

Differentiate both sides

$\mathbf{2A' = 12 - 6b}$

Set A' to 0

$\mathbf{2 \times 0 = 12 - 6b}$

$\mathbf{0 = 12 - 6b}$

Add 6b to both sides

$\mathbf{6b = 12}$

Divide both sides by 6

$\mathbf{b = 2}$

Substitute 2 for b in $\mathbf{A = \frac{12b - 3b^2}2}$

$\mathbf{A = \frac{12(2) - 3(2)^2}2}$

$\mathbf{A = \frac{24 - 12}2}$

$\mathbf{A = \frac{12}2}$

$\mathbf{A = 6}$

Hence, the largest area is 6 square yards

Read more about areas at:

https://brainly.com/question/1574916