Let's break down the problems one by one:

Question 7:

A cone and a cylinder have the same height $h$ and are equivalent in volume. Let $r$ represent the radius of the cylinder. We are to determine the radius of the cone.

Key points:

• Volume of a cylinder: $V_{\text{cylinder}} = \pi r^2 h$
• Volume of a cone: $V_{\text{cone}} = \frac{1}{3} \pi r_{\text{cone}}^2 h$

Since their volumes are equal: $\pi r^2 h = \frac{1}{3} \pi r_{\text{cone}}^2 h$

From this, we can solve for the radius $r_{\text{cone}}$ of the cone.

Question 8:

The rectangular prism and cube on the right are equivalent in volume. By how much does the total area of the prism surpass the total area of the cube?

Key points:

• Cube side length is $x$, so the surface area of the cube is $6x^2$.
• Rectangular prism dimensions are $(x-2)$, $x+4$, and $x$, so its surface area is calculated as: $\text{Area of prism} = 2[(x-2)(x+4) + (x-2)x + (x+4)x]$ Now, subtract the cube's surface area from the prism's surface area to find the difference.

Question 9:

A prism, cylinder, and cone are equivalent in volume. If the bases are equivalent:

1. Compare the heights of the prism and the cylinder.
2. Compare the heights of the prism and the cone.

Question 10:

A sculpture is formed by a pyramid mounted on top of a cube, with the volumes of the cube and pyramid equivalent. The total volume of the sculpture is 432 cm³. We are to find the total height of the sculpture.

Key points:

• Volume of cube: $V_{\text{cube}} = s^3$
• Volume of pyramid: $V_{\text{pyramid}} = \frac{1}{3} s^2 h_{\text{pyramid}}$

Since the total volume of the sculpture is known, we can express the total volume as the sum of the cube and pyramid volumes: $V_{\text{cube}} + V_{\text{pyramid}} = 432 \text{ cm}^3$

Now solve for the side length $s$ and total height $h_{\text{total}}$.

Let me know which part you'd like to focus on first or if you'd like further clarification! Here's how you can expand your understanding:

1. How are the volumes of cylinders and cones calculated in other scenarios?
2. How does the surface area change for various 3D shapes?
3. How are the relationships between height and volume compared for different shapes?
4. How does symmetry play a role in shape comparison problems?
5. What other examples can you think of where two shapes have equivalent volumes?

Tip: Always check if units are consistent when calculating areas and volumes to avoid mistakes.