A dog trainer has 64 ft of fencing that will be used to create a rectangular work area for dogs. If the trainer wants to enclose an area of 192 ft2, what will be the dimensions of the work area?

guessing way:

192 = 2^6*3 = 16*12 = 8*24
2(8+24) = 64
so the work area is 8x24

algebraic way:
If the pen's width is w, then the length is 32-w

w(32-w) = 192
w^2 - 32w + 192 = 0
(w-8)(w-24) = 0
w = 8 or 24
so, the pen is 8x24

Unless you use the quadratic formula, factoring the polynomial is essentially the same as guessing the right factors of 192!

Well, if the dog trainer has 64 ft of fencing to work with, they must really be in a "ruff" situation! But don't worry, I've got the "pawfect" solution for you.

Let's assume the length of the rectangular work area is L and the width is W. Since there are four sides to the rectangle, we can calculate the total length of the fencing using the equation:

2L + 2W = 64 ft

Now, the area of a rectangle is given by the formula A = L * W. We know that the area of the work area is 192 ft²:

L * W = 192 ft²

To solve for the dimensions, we can use the fact that 2L + 2W = 64 ft. We can rearrange the equation to solve for L:

2L = 64 ft - 2W
L = (64 ft - 2W) / 2

Substituting L in the area equation:

(64 ft - 2W) / 2 * W = 192 ft²
64 ft - 2W * W = 384 ft²
-2W² + 64 ft = 384 ft²
2W² = 64 ft - 384 ft²
W² = (64 ft - 384 ft²) / (-2)
W² = -320 ft² / -2
W² = 160 ft²
W = √160 ft
W ≈ 12.65 ft

Now, substituting the value of W back into the equation for L:

L = (64 ft - 2 * 12.65 ft) / 2
L = (64 ft - 25.3 ft) / 2
L = 38.7 ft / 2
L ≈ 19.35 ft

So, the dimensions of the work area will be approximately 19.35 ft by 12.65 ft. "Pawsome," right?

To find the dimensions of the work area, we can use the formula for the area of a rectangle: A = length * width.

Let's assume the length of the rectangle is L and the width is W.

According to the problem, the perimeter of the rectangle (which is twice the sum of the length and the width) is 64 ft.

So we have the equation: 2L + 2W = 64 ft.

Simplifying this equation, we get: L + W = 32.

We also know that the area of the rectangle is 192 ft^2, which gives us another equation: L * W = 192 ft^2.

To solve these two equations, we can use substitution or elimination method.

Let's solve it using the substitution method:

From the equation L + W = 32, we have W = 32 - L.
Substitute this value of W into the equation L * W = 192:
L * (32 - L) = 192.

Expanding this equation, we get 32L - L^2 = 192.

Rearranging terms, we obtain the quadratic equation: L^2 - 32L + 192 = 0.

To solve this equation, we can factor it or use the quadratic formula.

By factoring, we find (L - 12)(L - 16) = 0.

Setting each factor equal to zero, we get L - 12 = 0 or L - 16 = 0.

Solving these equations, we have L = 12 or L = 16.

Substituting these values back into the equation L + W = 32, we can find the corresponding widths:

For L = 12, W = 32 - L = 20.

For L = 16, W = 32 - L = 16.

Therefore, the possible dimensions of the work area are:

Length: 12 ft, Width: 20 ft
Length: 16 ft, Width: 16 ft

To find the dimensions of the work area, we can use the formulas for the perimeter and area of a rectangle.

Let's assume that the length of the rectangle is L and the width is W.

The formula for the perimeter is given by:
Perimeter = 2 * (Length + Width)

And the formula for the area is given by:
Area = Length * Width

Given that the perimeter is 64 ft, we can write the equation as:
64 ft = 2 * (L + W)

Simplifying further, we have:
32 ft = L + W

Given that the area is 192 ft^2, we can write the equation as:
192 ft^2 = L * W

Now, we have a system of two equations with two unknowns. We can solve this system to find the values of L and W.

From the first equation, we can rewrite it as:
L = 32 ft - W

Substituting this value of L into the second equation, we get:
192 ft^2 = (32 ft - W) * W

Expanding and rearranging the equation, we have:
192 ft^2 = 32 ft * W - W^2

Rearranging further, we have a quadratic equation:
W^2 - 32 ft * W + 192 ft^2 = 0

To solve this quadratic equation, we can use the quadratic formula:
W = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -32 ft, and c = 192 ft^2. Substituting these values into the quadratic formula, we get:
W = (-(-32 ft) ± √((-32 ft)^2 - 4 * 1 * 192 ft^2)) / (2 * 1)

Simplifying further, we have:
W = (32 ft ± √(1024 ft^2 - 768 ft^2)) / 2

W = (32 ft ± √256 ft^2) / 2

W = (32 ft ± 16 ft) / 2

Now, we have two possible values for W:
1. W1 = (32 ft + 16 ft) / 2 = 48 ft / 2 = 24 ft
2. W2 = (32 ft - 16 ft) / 2 = 16 ft / 2 = 8 ft

Substituting each value of W back into the equation L = 32 ft - W, we can find the corresponding values of L:
1. L1 = 32 ft - 24 ft = 8 ft
2. L2 = 32 ft - 8 ft = 24 ft

Therefore, the dimensions of the work area can be either 8 ft by 24 ft or 24 ft by 8 ft.