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Lubov Fominskaja [6]
2 years ago
9

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

Mathematics
1 answer:
marysya [2.9K]2 years ago
3 0

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

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