Question

1. a. find the perimeter of the interior rectangle.

b. find the area of the shaded region.

Explanation:

Step1: Calculate the perimeter of the interior rectangle

The formula for the perimeter of a rectangle is \(P = 2(l + w)\), where \(l=2x + 7\) and \(w=x + 1\).

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Step2: Calculate the area of the shaded region

The area of the shaded region \(A=\text{Area of the outer rectangle}-\text{Area of the inner rectangle}\).
The area of a rectangle is \(A = lw\). For the outer rectangle, \(l = 4x + 5\) and \(w=3x - 2\), so \(A_{outer}=(4x + 5)(3x - 2)\).

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For the inner rectangle, \(l = 2x + 7\) and \(w=x + 1\), so \(A_{inner}=(2x + 7)(x + 1)\).

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Then \(A=(12x^{2}+7x - 10)-(2x^{2}+9x + 7)\).

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Answer:

a. The perimeter of the interior rectangle is \(6x + 16\).
b. The area of the shaded region is \(10x^{2}-2x - 17\).