What kind of shape does the graph for 3rsin(theta)+5r^2 cos^2(theta) = 2 make?

To analyze the shape of the graph represented by the polar equation 

\[
3r \sin(\theta) + 5r^2 \cos^2(\theta) = 2
\]

we can try rewriting it in a more familiar form or investigating its key properties.

### Step 1: Simplify the Equation
Rewrite the equation in terms of \( r \) and trigonometric identities:
\[
3r \sin(\theta) + 5r^2 \cos^2(\theta) = 2.
\]
Notice that the second term, \( 5r^2 \cos^2(\theta) \), could be more manageable if we change it to \( r^2 \cos^2(\theta) = (r \cos(\theta))^2 \). This substitution helps us because \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) in Cartesian coordinates.

### Step 2: Convert to Cartesian Coordinates
Using \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), the terms translate as follows:
- \( r \sin(\theta) = y \)
- \( r \cos(\theta) = x \), thus \( r^2 \cos^2(\theta) = x^2 \).

Substitute these into the equation:
\[
3y + 5x^2 = 2.
\]

### Step 3: Rearrange the Equation
Now, we have the equation in Cartesian form:
\[
5x^2 + 3y = 2.
\]
Solving for \( y \) gives:
\[
y = \frac{2 - 5x^2}{3}.
\]

### Step 4: Analyze the Shape
This equation \( y = \frac{2 - 5x^2}{3} \) resembles the form of a **parabola**. The term \( -5x^2 \) indicates that it opens downward along the \( y \)-axis. Hence, the graph will be a parabola opening downward with a vertex at \( \left(0, \frac{2}{3}\right) \).

### Conclusion
The graph of the given polar equation represents a **downward-opening parabola**.

Would you like further details on the derivation, or have any questions?

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Here are some related questions for further exploration:
1. How can we determine if a polar equation represents a conic section?
2. What effect do coefficients have on the shape of a polar graph?
3. How would the equation change if we replaced \( \cos^2(\theta) \) with \( \sin^2(\theta) \)?
4. How can we convert between polar and Cartesian coordinates systematically?
5. What other methods can be used to analyze the shape of polar equations?

**Tip:** Converting polar equations to Cartesian form can simplify analysis by transforming unfamiliar terms into more recognizable ones.

What kind of shape does the graph for 3r sin(theta) + 5r^2 cos^2(theta) = 2 make?

Explore the shape of the polar equation 3r sin(theta) + 5r^2 cos^2(theta) = 2, converted to a Cartesian form as a downward-opening parabola. Includes step-by-step simplification and conversion methods for students in Grades 10-12.

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