Yes please continue, we are looking for the height of the cone as well as the height of the cylinder.

Let's continue by solving the quadratic equation for the height of the cone \( h \).

### Step 1: Solve the Quadratic Equation

The quadratic equation we have is:

\[
63h^2 - 1285.28h + 6309.28 = 0
\]

This equation is in the standard quadratic form \( ax^2 + bx + c = 0 \), where:
- \( a = 63 \)
- \( b = -1285.28 \)
- \( c = 6309.28 \)

We can solve this using the quadratic formula:

\[
h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values for \( a \), \( b \), and \( c \):

\[
h = \frac{-(-1285.28) \pm \sqrt{(-1285.28)^2 - 4 \cdot 63 \cdot 6309.28}}{2 \cdot 63}
\]

Now, calculate the discriminant:

\[
b^2 - 4ac = (-1285.28)^2 - 4 \cdot 63 \cdot 6309.28
\]

Let's compute these values step-by-step.The solutions for the height of the cone \(h\) are:

\[
h_1 \approx 12.18 \text{ units}, \quad h_2 \approx 8.22 \text{ units}
\]

Since both values are positive, we need to determine which one fits our scenario. Given that the total surface area is a specific number (964π), either height can be correct in different contexts. However, typically, we consider the smaller height as it keeps the values more reasonable.

Let's select \(h = 8.22\) units as the height of the cone.

### Step 2: Calculate the Height of the Cylinder

Since the height of the cylinder is four times the height of the cone:

\[
H = 4h = 4 \times 8.22 = 32.88 \text{ units}
\]

### Final Answer:
- **Height of the Cone**: \(h = 8.22\) units
- **Height of the Cylinder**: \(H = 32.88\) units

Would you like to verify this with the original surface area, or do you have any other questions? Here are 5 questions to consider:

1. What would be the surface area if the height of the cone was \(12.18\) units?
2. How does changing the radius affect the height calculations?
3. What are the implications if the cylinder height is not a simple multiple of the cone height?
4. How do you derive the formula for the lateral surface area of a cone?
5. Can this method be generalized for different composite shapes?

**Tip:** Remember to double-check calculations for composite shapes by ensuring all surface area components are included.

Learn how to calculate the heights of a cone and cylinder given a total surface area of 964π square units. This problem involves solving a quadratic equation for the height of the cone and understanding the relationship between the cone and cylinder heights. Suitable for high school students studying geometry and algebra.

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