Explore a vast range of topics and get informed answers at IDNLearn.com. Whether your question is simple or complex, our community is here to provide detailed and trustworthy answers quickly and effectively.
Sagot :
To find the area of the region bounded by the given ellipse [tex]\( \frac{x^2}{64} + \frac{y^2}{100} = 1 \)[/tex], we need to follow these steps:
1. Understand the standard form of the ellipse equation:
The standard form of an ellipse equation is:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
2. Identify [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] from the given equation:
From the given equation [tex]\( \frac{x^2}{64} + \frac{y^2}{100} = 1 \)[/tex]:
- [tex]\( a^2 = 64 \)[/tex]
- [tex]\( b^2 = 100 \)[/tex]
3. Calculate [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
To find [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we take the square root of [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex]:
[tex]\[ a = \sqrt{64} = 8 \][/tex]
[tex]\[ b = \sqrt{100} = 10 \][/tex]
4. Use the formula for the area of an ellipse:
The area [tex]\( A \)[/tex] of an ellipse is given by the formula:
[tex]\[ A = \pi \cdot a \cdot b \][/tex]
5. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ A = \pi \cdot 8 \cdot 10 \][/tex]
6. Calculate the area:
[tex]\[ A = 80\pi \][/tex]
Therefore, the area of the region bounded by the ellipse is:
[tex]\[ \boxed{80 \pi \text{ sq. units}} \][/tex]
Thus, the correct option is (B) [tex]\( 80 \pi \)[/tex].
1. Understand the standard form of the ellipse equation:
The standard form of an ellipse equation is:
[tex]\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \][/tex]
2. Identify [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex] from the given equation:
From the given equation [tex]\( \frac{x^2}{64} + \frac{y^2}{100} = 1 \)[/tex]:
- [tex]\( a^2 = 64 \)[/tex]
- [tex]\( b^2 = 100 \)[/tex]
3. Calculate [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
To find [tex]\( a \)[/tex] and [tex]\( b \)[/tex], we take the square root of [tex]\( a^2 \)[/tex] and [tex]\( b^2 \)[/tex]:
[tex]\[ a = \sqrt{64} = 8 \][/tex]
[tex]\[ b = \sqrt{100} = 10 \][/tex]
4. Use the formula for the area of an ellipse:
The area [tex]\( A \)[/tex] of an ellipse is given by the formula:
[tex]\[ A = \pi \cdot a \cdot b \][/tex]
5. Substitute [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[ A = \pi \cdot 8 \cdot 10 \][/tex]
6. Calculate the area:
[tex]\[ A = 80\pi \][/tex]
Therefore, the area of the region bounded by the ellipse is:
[tex]\[ \boxed{80 \pi \text{ sq. units}} \][/tex]
Thus, the correct option is (B) [tex]\( 80 \pi \)[/tex].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. IDNLearn.com is dedicated to providing accurate answers. Thank you for visiting, and see you next time for more solutions.