Question

A rectangular fence with a positive area has length represented by the expression 3x2+5x-8 and width by 2x2+6x

Answer 1

Answer:

1. $Perimeter = 10x^2+ 22x - 16$

2. $Area = 6x^4 +28x^3 + 14x^2 - 48x$

3. $x = 1$ or $x = -3.2$

4. $Length = 0$ and $Width = 8$

5. $Length = 6.72$ and $Width = 1.28$

6. Second Solution

7. $Area = 8.6016$

Step-by-step explanation:

Given

$Length = 3x^2 + 5x - 8$

$Width = 2x^2 + 6x$

Solving (1): The perimeter

Perimeter is calculated as thus:

$Perimeter = 2 * (Length + Width)$

Substitute values for Length and Width

$Perimeter = 2 * (3x^2 + 5x - 8 + 2x^2 + 6x)$

Collect Like Terms

$Perimeter = 2 * (3x^2+ 2x^2 + 6x + 5x - 8 )$

$Perimeter = 2 * (5x^2+ 11x - 8 )$

Open Bracket

$Perimeter = 10x^2+ 22x - 16$

Solving (2): The Area:

Area is calculated as thus:

$Area = Length * Width$

Substitute values for Length and Width

$Area = (3x^2 + 5x - 8) * (2x^2 + 6x)$

Expand Bracket

$Area = 2x^2(3x^2 + 5x - 8) + 6x(3x^2 + 5x - 8)$

Open Bracket

$Area = 6x^4 + 10x^3 - 16x^2 + 18x^3 + 30x^2 - 48x$

Collect Like Terms

$Area = 6x^4 + 10x^3 + 18x^3 - 16x^2 + 30x^2 - 48x$

$Area = 6x^4 +28x^3 + 14x^2 - 48x$

Solving (3): If perimeter is 16, show that x = 1 and x = -3.2

In (a) $Perimeter = 10x^2+ 22x - 16$

Substitute 16 for Perimeter

$16 = 10x^2+ 22x - 16$

Equate to 0

$10x^2+ 22x - 16 -16=0$

$10x^2+ 22x - 32=0$

Expand

$10x^2+ 32x -10x- 32=0$

Factorize

$x(10x + 32) - 1(10x + 32) = 0$

$(x - 1)(10x + 32) = 0$

Split

$x - 1 =0$ or $10x + 32 = 0$

$x = 1$ or $10x = -32$

$x = 1$ or $x = -32/10$

$x = 1$ or $x = -3.2$

Solving (4): Using x = 1; Solve the dimension of the rectangle

We have that:

$Length = 3x^2 + 5x - 8$

$Width = 2x^2 + 6x$

Substitute 1 for x in the given parameters

$Length = 3(1)^2 + 5(1) - 8$

$Length = 3*1 + 5*1 - 8$

$Length = 3 + 5 - 8$

$Length = 0$

$Width = 2(1)^2 + 6(1)$

$Width = 2*1 + 6*1$

$Width = 2 + 6$

$Width = 8$

Solving (5): Using x = -3.2; Solve the dimension of the rectangle

We have that:

$Length = 3x^2 + 5x - 8$

$Width = 2x^2 + 6x$

Substitute -3.2 for x in the given parameters

$Length = 3(-3.2)^2 + 5(-3.2) - 8$

$Length = 3*10.24 + 5(-3.2) - 8$

$Length = 30.72 -16 - 8$

$Length = 6.72$

$Width = 2(-3.2)^2 + 6(-3.2)$

$Width = 2*10.24 + 6(-3.2)$

$Width = 20.48 -19.2$

$Width = 1.28$

$Length = 6.72$ and $Width = 1.28$

Solving (6):

In the first solution of the dimensions in number 4, we have that

$Length = 0$ and $Width = 8$

This dimension can not be considered because the length of a rectangle can not be 0

In the second solution of the dimensions in number 5, we have that

$Length = 6.72$ and $Width = 1.28$

This dimension makes more sense because both dimensions are greater than 0

Solving (7): Using (6), determine the Area

In (6), we conclude that

$Length = 6.72$ and $Width = 1.28$

So, Area is calculated as thus:

$Area = Length * Width$

$Area = 6.72 * 1.28$

$Area = 8.6016$