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Find the area of a parallelogram with sides of 6 feet and 10 feet, given that one of the angles is 45°.

A. [tex]\(30\sqrt{2} \, \text{ft}^2\)[/tex]
B. 30 [tex]\(\text{ft}^2\)[/tex]
C. 60 [tex]\(\text{ft}^2\)[/tex]


Sagot :

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].
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