Let's analyze the problem from the uploaded image:

We are tasked with finding the missing dimension (a binomial) for the rectangular prism on the right, given that both prisms have the same volume.

Step 1: Volume of the Hexagonal Prism (Left Prism)

The volume of the left prism is calculated as: $\text{Volume} = \text{Area of Base} \times \text{Height}$

Base Area of the Hexagonal Prism:

The base of the hexagonal prism consists of a regular hexagon divided into six equilateral triangles.

Each equilateral triangle has a side length of $6b$ and an area formula: $\text{Area of one triangle} = \frac{\sqrt{3}}{4} \times (\text{side})^2$

So, the total area of the hexagon is: $\text{Area of Hexagon} = 6 \times \frac{\sqrt{3}}{4} \times (6b)^2 = 6 \times \frac{\sqrt{3}}{4} \times 36b^2$ $\text{Area of Hexagon} = 54\sqrt{3}b^2$

Height of the Hexagonal Prism:

The height is given as $8 \, \text{cm}$.

Total Volume of the Hexagonal Prism:

$\text{Volume}_{\text{Hexagonal Prism}} = 54\sqrt{3}b^2 \times 8 = 432\sqrt{3}b^2$

Step 2: Volume of the Rectangular Prism (Right Prism)

The volume of the rectangular prism is also given as: $\text{Volume}_{\text{Rectangular Prism}} = \text{Base Area} \times \text{Height}$

Base Area of the Rectangular Prism:

The base area is: $\text{Base Area} = 5 \times (4b) = 20b$

Total Volume of the Rectangular Prism:

The height of the rectangular prism is unknown, represented as $?$. Let this height be $x$. Then: $\text{Volume}_{\text{Rectangular Prism}} = 20b \times x = 20bx$

Step 3: Equating the Volumes

Since the volumes of both prisms are equal: $432\sqrt{3}b^2 = 20bx$

Solve for $x$: $x = \frac{432\sqrt{3}b^2}{20b}$ $x = \frac{432\sqrt{3}b}{20}$ $x = 21.6\sqrt{3}b$

Step 4: Representing the Missing Dimension

The missing height of the rectangular prism is: $x = 21.6\sqrt{3}b$

Would you like a detailed explanation of any of the steps or further simplification? 😊

Related Questions:

1. How is the volume of a hexagonal prism derived in general?
2. Why does the base of a hexagonal prism consist of six equilateral triangles?
3. What is the relationship between the area of a hexagon and its side length?
4. How would the solution change if one of the $b$-values were different?
5. Could you calculate the exact numerical value of the missing height for $b = 1$?

Tip:

Always verify that units and terms cancel correctly when equating expressions—this ensures your algebraic manipulations are valid.