To find the area of a parallelogram with given side lengths and one of the angles, we can use the formula:
[tex]\[ \text{Area} = a \cdot b \cdot \sin(\theta) \][/tex]
Where:
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the lengths of the sides of the parallelogram.
- [tex]\(\theta\)[/tex] is the included angle between those sides in degrees.
Step-by-Step Solution:
1. Identify the given values:
- Side [tex]\(a = 6\)[/tex] feet
- Side [tex]\(b = 10\)[/tex] feet
- Included angle [tex]\(\theta = 45^\circ\)[/tex]
2. Convert the angle from degrees to radians:
- We know that [tex]\(1\)[/tex] degree = [tex]\(\frac{\pi}{180}\)[/tex] radians.
- Therefore, [tex]\(45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4}\)[/tex] radians.
3. Calculate the sine of the angle:
- [tex]\(\sin(45^\circ) = \sin\left(\frac{\pi}{4}\right) \approx 0.7071\)[/tex]
4. Apply the area formula:
[tex]\[
\text{Area} = a \cdot b \cdot \sin(\theta)
\][/tex]
Plugging in the values:
[tex]\[
\text{Area} = 6 \text{ feet} \cdot 10 \text{ feet} \cdot \sin\left(\frac{\pi}{4}\right)
\][/tex]
[tex]\[
\text{Area} = 6 \text{ feet} \cdot 10 \text{ feet} \cdot 0.7071
\][/tex]
[tex]\[
\text{Area} = 42.4264 \text{ square feet}
\][/tex]
So, the area of the parallelogram is approximately [tex]\(42.4264\)[/tex] square feet.
Given the options, none of the exact given results match directly [tex]\(42.4264\)[/tex] square feet, but an approximate comparison shows that it's closest to [tex]\(30\sqrt{2}\)[/tex] ft² since:
[tex]\[ 30\sqrt{2} \approx 30 \times 1.4142 = 42.426 \][/tex]
Thus, the correct answer is [tex]\( \boxed{30\sqrt{2}} \text{ ft}^2 \)[/tex].
