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].
