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To solve \( f(x, y) = \sqrt{16 - x^2 - 16y^2} \) using the **slice method**, we'll look at slices of the graph in specific planes, particularly fixing either \(x\) or \(y\) and analyzing how the graph behaves.

### Step-by-Step Solution:

The equation can be rewritten as:
\[
f(x, y) = \sqrt{16 - x^2 - 16y^2}
\]
This is the upper half of an elliptical surface, since we are taking the square root, which results in non-negative \( z \)-values.

### 1. **Slice along \( x = 0 \):**
When \( x = 0 \), the equation simplifies to:
\[
f(0, y) = \sqrt{16 - 16y^2} = 4\sqrt{1 - y^2}
\]
This describes a semicircle in the \( yz \)-plane, with a radius of 1 for \( y \). The graph at \( x = 0 \) is a semicircle of radius 1.

### 2. **Slice along \( y = 0 \):**
When \( y = 0 \), the equation simplifies to:
\[
f(x, 0) = \sqrt{16 - x^2}
\]
This describes a semicircle in the \( xz \)-plane, with a radius of 4. The graph at \( y = 0 \) is a semicircle of radius 4.

### 3. **Slice along constant \( y \) values:**
For general constant values of \( y = c \), the equation becomes:
\[
f(x, c) = \sqrt{16 - x^2 - 16c^2}
\]
This is again a semicircle in the \( xz \)-plane, but with a decreasing radius. The radius is \( \sqrt{16 - 16c^2} = 4\sqrt{1 - c^2} \), which means the radius of the semicircle depends on \( c \), shrinking as \( |c| \) increases from 0 to 1. When \( |y| = 1 \), the graph collapses to a point.

### 4. **Graph Description in Words:**
The surface described by \( f(x, y) \) is an **elliptical dome**. At \( x = 0 \), the cross-section is a semicircle of radius 1. As you move along the \( x \)-axis, the cross-section becomes a semicircle with a radius of 4 when \( y = 0 \). The graph smoothly transitions, shrinking along the \( y \)-axis until it converges to a point at \( y = \pm 1 \).

In 3D, the graph is an **ellipsoid** with the following key properties:
- It stretches farther along the \( x \)-axis, with a maximum radius of 4.
- Along the \( y \)-axis, the ellipsoid has a radius of 1.

Would you like any specific details or clarifications on this? Below are a few related questions for further understanding:

### 5 Related Questions:
1. How do you derive the equation of an ellipsoid from a given equation?
2. What happens to the graph if we take the negative square root of the equation?
3. How would the graph change if the coefficient of \( y^2 \) was different?
4. What is the volume enclosed by this surface?
5. How would the graph look if we didn't take the square root (i.e., used \( z^2 = 16 - x^2 - 16y^2 \))?

### Tip:
When using the slice method, it is helpful to visualize each cross-section of the surface in 2D and then mentally combine them to build the full 3D shape.

Sketch a graph of f(x, y) = √(16 - x^2 - 16y^2). Describe your graph in words.

Explore the detailed steps in sketching the graph of the function f(x, y) = √(16 - x^2 - 16y^2), representing the upper half of an elliptic surface. Learn about cross-sectional slicing and visualizing functions in 3D space. Suitable for college-level students in multivariable calculus.

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