1. The length of a rectangle is 6 inches longer than thrice its width. The area of the rectangle is 57 square inches. Write a quadratic equation for the area of the floor in terms of x. Find the length and width of the rectangle.

2. If 1,500 ft of fencing is available to build and fence ten adjacent pens, express the total area of the pens in terms of x. What value of x and y will maximize the total area? Find the total area of the field.

width --- x

length -- 3x + 6

x(3x+6) = 57
3x^2 + 6x - 57 = 0
x^2 + 2x - 19 = 0
I am going to complete the square, easier in this case than the formula
x^2 + 2x + 1 = 19+1
(x+1)^2 = 20
x+1 = ± √20
x = -1 ± √20
= appr 3.472 or a negative

width = 3.472 , lenght is 16.416

Perhaps you were supposed to use 2 variables
then width = x
length = y

but y = 3x+6 , and area = xy = x(3x+6)
and we are back to the way I did it at the top

2.
Your "English" of the first sentence makes no sense to me. How is x defined? Where does y come in?
Tell me what your definitions are.

1. To find the length and width of the rectangle, we can start by expressing the information given as equations. Let's say the width of the rectangle is x inches.

According to the problem, the length of the rectangle is 6 inches longer than thrice its width. Hence, the length can be expressed as (3x + 6) inches.

The area of a rectangle is calculated by multiplying its length and width. In this case, the area is given as 57 square inches. Therefore, we can create the equation:

Area = Length * Width
57 = (3x + 6) * x

To solve for x, we first need to expand the equation:
57 = 3x^2 + 6x

Now, let's rearrange the equation and set it equal to zero to form a quadratic equation:

3x^2 + 6x - 57 = 0

This is the quadratic equation that represents the area of the rectangle in terms of x. To find the length and width, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.

Once you solve the quadratic equation, you will obtain the values for x, which represents the width. Then, you can substitute this value into the equation (3x + 6) to find the length.

2. To express the total area of the pens in terms of x, let's assume the width of each pen is x feet.

Since there are ten adjacent pens, the total length of the fencing used will be equal to 10 times the width of each pen. Hence, the total length of the fencing is 10x feet.

The perimeter of each pen is calculated by adding the width and length of the pen. Therefore, the total fencing needed for each pen is 2 times the width plus 2 times the length, which can be expressed as 2x + 2(10x) = 2x + 20x = 22x feet.

According to the problem, only 1,500 ft of fencing is available. Therefore, we have the equation:

22x = 1500

Now, to find the total area, we need to determine the product of the width and length of each pen, multiplied by the total number of pens (10 in this case). This can be expressed as:

Total Area = width * length * number of pens
Total Area = (x * 10x * 10) square feet
Total Area = 100x^2 square feet

To find the value of x and y that maximizes the total area, we need more information or constraints about the relationship between x and y. Without that additional information, it is not possible to provide a specific value for x or y.

To find the maximum total area, differentiate the equation 100x^2 with respect to x, set the derivative equal to zero, and solve for x.