To find the surface area of an ellipsoid cap, we need to use the formula specific to the cap of an ellipsoid, which depends on the dimensions of the ellipsoid and the height of the cap.

Given the ellipsoid with semi-axes $a = 8$, $b = 6$, and $c = 5$, and the height $h = 3$, we can proceed with the following steps.

Formula for Ellipsoid Cap Surface Area

For a general ellipsoid, the surface area of the cap can be approximated using:

$A_{\text{cap}} \approx 2\pi \cdot a \cdot b \left( 1 - \frac{h}{c} \right)$

Where:

• $a, b, c$ are the semi-axes of the ellipsoid.
• $h$ is the height of the cap.

Now, substituting the given values:

• $a = 8$
• $b = 6$
• $c = 5$
• $h = 3$

Substituting these into the formula:

$A_{\text{cap}} \approx 2\pi \cdot 8 \cdot 6 \left( 1 - \frac{3}{5} \right)$

$A_{\text{cap}} \approx 2\pi \cdot 8 \cdot 6 \left( \frac{2}{5} \right)$

$A_{\text{cap}} \approx 2\pi \cdot 48 \cdot \frac{2}{5}$

$A_{\text{cap}} \approx 2\pi \cdot 19.2$

$A_{\text{cap}} \approx 38.4\pi$

$A_{\text{cap}} \approx 120.6 \, \text{square units}$

So, the surface area of the ellipsoid cap is approximately 120.6 square units.

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Would you like further details on the formula or any related concepts? Here are some related questions for you:

1. How do you calculate the full surface area of an ellipsoid?
2. What is the relationship between the surface area of an ellipsoid cap and the height of the cap?
3. How does the shape of an ellipsoid affect the surface area of a cap?
4. How can you calculate the volume of an ellipsoid cap?
5. What changes would occur to the surface area if $h$ was larger or smaller?

Tip: When dealing with ellipsoid-related calculations, visualize how the semi-axes interact—changing one value (like $a$) significantly alters the shape and surface area.