Question: User: Two resistors wired in parallel in an electrical circuit give a effective resistance of
R(x,y) = xy / (x + y) where x and y are the positive resistances of the individual resistors
(typically measured in ohms). Complete parts a through c below.
Part 1 of 3
a. Graph the resistance function using the window [0,10] \u00D7 [0,10] \u00D7 [0,5]. Choose the correct graph below.
A.
B.
C.
Graph shape: a 3D surface patch rising from the origin into the positive x, positive y, and positive z region; it looks like a curved slanted sheet, with the correct choice appearing to be option C. position_hint = center
1. **Problem Statement:**
We are given the effective resistance function for two resistors wired in parallel:
$$R(x,y) = \frac{xy}{x + y}$$
where $x$ and $y$ are positive resistances.
We need to graph this function over the domain $x \in [0,10]$, $y \in [0,10]$, and range $R(x,y) \in [0,5]$.
2. **Understanding the function:**
- The formula for parallel resistors is:
$$R = \frac{xy}{x + y}$$
- Both $x$ and $y$ are positive, so the function is defined and positive.
- When either $x$ or $y$ approaches zero, $R$ approaches zero.
- When $x = y$, the resistance is:
$$R = \frac{x \cdot x}{x + x} = \frac{x^2}{2x} = \frac{x}{2}$$
which increases linearly with $x$.
3. **Graph shape analysis:**
- The surface starts at the origin $(0,0,0)$.
- It rises smoothly as $x$ and $y$ increase.
- The surface is curved and slanted upwards in the positive $x$, $y$, and $z$ directions.
- The maximum value in the given window is capped at $5$.
4. **Conclusion:**
The correct graph is option C, which shows a 3D surface patch rising from the origin into the positive $x$, $y$, and $z$ region, matching the shape of the function.
**Final answer:** Option C is the correct graph for the resistance function $R(x,y) = \frac{xy}{x + y}$ over the specified window.