Question

1. find the perimeter of a rectangle if the length is ( x^3 + 8x^2 - 2 ) and the width is ( 3x^3 - 9x ). (hint: the perimeter of a rectangle is given by the formula ( p = 2l + 2w ).)

Explanation:

Step1: Recall the perimeter formula

The perimeter \( P \) of a rectangle is given by \( P = 2l + 2w \), where \( l \) is the length and \( w \) is the width.

Step2: Substitute the given length and width

Given \( l = r^3 + 8r^2 - 2 \) and \( w = 3r^2 - 9r \). Substitute these into the formula:

$$
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$$

Step3: Distribute the 2s

Using the distributive property \( a(b + c)=ab + ac \):

$$
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$$

Step4: Combine like terms

Combine the \( r^2 \) terms: \( 16r^2+6r^2 = 22r^2 \)

$$ P = 2r^3+22r^2 - 18r - 4 $$

Answer:

The perimeter of the rectangle is \( 2r^3 + 22r^2 - 18r - 4 \)