Let's begin by identifying the dimensions of Mrs. Ishimitsu's coffee table given in the problem:
- The length of the table is [tex]\( 2x^2 - 16 \)[/tex] feet.
- The width of the table is [tex]\( -x^2 + 4x + 1 \)[/tex] feet.
To find the perimeter of a rectangle, we use the formula:
[tex]\[
\text{Perimeter} = 2 \times (\text{Length} + \text{Width})
\][/tex]
Now we substitute the given expressions for the length and the width into this formula:
[tex]\[
\text{Perimeter} = 2 \times \left( (2x^2 - 16) + (-x^2 + 4x + 1) \right)
\][/tex]
First, simplify the expression inside the parentheses:
[tex]\[
(2x^2 - 16) + (-x^2 + 4x + 1) = 2x^2 - x^2 + 4x - 16 + 1 = x^2 + 4x - 15
\][/tex]
Then multiply by 2:
[tex]\[
\text{Perimeter} = 2 \times (x^2 + 4x - 15) = 2x^2 + 8x - 30
\][/tex]
Therefore, the expression for the total perimeter of the table in terms of [tex]\( x \)[/tex] is [tex]\( 2x^2 + 8x - 30 \)[/tex].
Now, if [tex]\( x = 3 \)[/tex], we need to find the length of the entire rubber bumper using our expression for the perimeter.
Substitute [tex]\( x = 3 \)[/tex] into the perimeter formula [tex]\( 2x^2 + 8x - 30 \)[/tex]:
[tex]\[
\text{Perimeter} = 2(3)^2 + 8(3) - 30
\][/tex]
Calculate each term:
[tex]\[
2(9) + 24 - 30 = 18 + 24 - 30 = 12
\][/tex]
Thus, if [tex]\( x = 3 \)[/tex], the length of the entire rubber bumper is 12 feet.
The correct option is:
[tex]\[ 2x^2 + 8x - 30; 12 \text{ feet} \][/tex]
