Sure, let's work through this problem step-by-step.
1. Understand the dilation concept: Dilation changes the size of a shape but preserves the proportion of its sides. When a shape is dilated with a scale factor [tex]\( k \)[/tex], the area of the dilated shape scales by [tex]\( k^2 \)[/tex].
2. Identify the given data:
- Original area of the trapezoid, [tex]\( A_{\text{original}} = 36 \, \text{in}^2 \)[/tex]
- Scale factor, [tex]\( k = 0.8 \)[/tex]
3. Apply the area scaling factor: When a shape is scaled by a factor [tex]\( k \)[/tex], the new area [tex]\( A_{\text{new}} \)[/tex] can be calculated using the formula:
[tex]\[ A_{\text{new}} = A_{\text{original}} \times k^2 \][/tex]
4. Calculate the new area:
[tex]\[
A_{\text{new}} = 36 \, \text{in}^2 \times (0.8)^2
\][/tex]
5. Calculate [tex]\( (0.8)^2 \)[/tex]:
[tex]\[
(0.8)^2 = 0.64
\][/tex]
6. Multiply the original area by 0.64:
[tex]\[
A_{\text{new}} = 36 \, \text{in}^2 \times 0.64 = 23.04 \, \text{in}^2
\][/tex]
Therefore, the area of the new trapezoid after dilation is [tex]\( 23.04 \, \text{in}^2 \)[/tex].
Of the given options, the correct answer is:
[tex]\( 23.04 \, \text{in}^2 \)[/tex].
