Question

1. find the perimeter of the rectangle at the right. (remember perimeter comes from adding up all the sides)

1. find the area of the rectangle at the right. remember the area of a rectangle is a = l x w.

1. suppose you know that the area of a rectangle is 40 square inches. the area can be modeled using the expression ( n^2 - 6n ). find the dimensions of the rectangle.

Explanation:

Step1: Recall perimeter formula for rectangle

Perimeter $P = 2(\text{length} + \text{width})$

Step2: Substitute given side lengths

$P = 2[(2x-5) + (x+4)]$

Step3: Simplify inside the parentheses

$P = 2(3x - 1)$

Step4: Distribute the 2

$P = 6x - 2$

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Step1: Recall area formula for rectangle

Area $A = \text{length} \times \text{width}$

Step2: Substitute given side lengths

$A = (2x-5)(x+4)$

Step3: Expand using FOIL method

$A = 2x(x) + 2x(4) -5(x) -5(4)$

Step4: Simplify the terms

$A = 2x^2 + 8x -5x -20 = 2x^2 + 3x -20$

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Step1: Set area expression equal to 40

$n^2 - 6n = 40$

Step2: Rearrange to standard quadratic form

$n^2 - 6n - 40 = 0$

Step3: Factor the quadratic equation

$(n - 10)(n + 4) = 0$

Step4: Solve for $n$

$n - 10 = 0 \implies n=10$; $n+4=0 \implies n=-4$

Step5: Discard negative value (length can't be negative)

$n=10$

Step6: Find the two dimensions

First dimension: $n=10$ inches; Second dimension: $n-6=10-6=4$ inches

Answer:

1. $6x - 2$
2. $2x^2 + 3x - 20$
3. 4 inches and 10 inches