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### Part B: Height of a cone

You are asked to find the height \( h \) of a cone with a volume \( V = 32\pi \, \text{cm}^3 \) and a radius \( r = 4 \, \text{cm} \).

The formula for the volume \( V \) of a cone is:

\[
V = \frac{1}{3} \pi r^2 h
\]

Given the volume \( 32\pi \) and radius \( 4 \, \text{cm} \), plug these values into the formula:

\[
32\pi = \frac{1}{3} \pi (4)^2 h
\]

Now, solve for \( h \):

1. First, cancel out \( \pi \) from both sides:

\[
32 = \frac{1}{3} (16) h
\]

2. Multiply both sides by 3 to eliminate the fraction:

\[
96 = 16h
\]

3. Divide both sides by 16:

\[
h = \frac{96}{16} = 6 \, \text{cm}
\]

So, the height of the cone is \( 6 \, \text{cm} \).

---

### Part 10: Solving the inequality

You are given the inequality:

\[
4(x + 3) - 7 \geq x + 3(x + 1)
\]

Let's simplify and solve step-by-step:

1. Expand both sides:

\[
4x + 12 - 7 \geq x + 3x + 3
\]

This simplifies to:

\[
4x + 5 \geq 4x + 3
\]

2. Subtract \( 4x \) from both sides:

\[
5 \geq 3
\]

Since this is always true, the inequality holds for all real numbers.

Thus, the solution is **all real numbers**.

---

Would you like further details or clarifications on these solutions?

Here are 5 related questions to explore:
1. How do you find the volume of a cylinder if you know the height and radius?
2. How does the formula for the volume of a sphere differ from that of a cone?
3. How would the volume change if the radius of the cone was doubled?
4. What are the general steps to solve linear inequalities?
5. How do inequalities change when both sides contain a variable?

**Tip:** When solving inequalities, always check whether you need to reverse the inequality symbol, especially when multiplying or dividing by negative numbers!

Part B: Find the height h of a cone with volume V = 32π cm³ and radius r = 4 cm. Also, solve the inequality: 4(x + 3) - 7 ≥ x + 3(x + 1).

Learn how to calculate the height of a cone with a given volume of 32π cm³ and a radius of 4 cm, using the volume formula for cones. Additionally, solve a linear inequality 4(x + 3) - 7 ≥ x + 3(x + 1) step-by-step. Suitable for Grades 8-10 students.

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