The problem provided involves a square-based pyramid. The given information includes:

• The ratio of the slant height ($l$) to the side length of the square base ($a$) is $5:6$.
• The total surface area of the pyramid is $1536 \, \text{cm}^2$.

You are asked to write the relation among the base area ($A$), the height ($h$), and the volume ($V$) of the pyramid.

Step 1: Relation Among Base Area, Height, and Volume

1. Base Area ($A$): For a square base with side length $a$, the base area is given by: $A = a^2$

2. Height ($h$): The height $h$ of the pyramid relates to the slant height $l$ and the base side length $a$ using the Pythagorean theorem. Since the height, half of the base ($a/2$), and slant height form a right triangle: $h^2 + \left(\frac{a}{2}\right)^2 = l^2$

3. Volume ($V$): The volume of a pyramid is given by: $V = \frac{1}{3} A h = \frac{1}{3} a^2 h$

Step 2: Given Ratios and Surface Area

• The ratio $l:a = 5:6$ implies $l = \frac{5}{6}a$.
• The total surface area is the sum of the base area and the area of the four triangular faces: $\text{Surface Area} = A + 4 \times \text{Area of one triangular face}$ The area of one triangular face (with base $a$ and height $l$) is: $\frac{1}{2} a l = \frac{1}{2} a \times \frac{5}{6}a = \frac{5}{12} a^2$ Thus, the total surface area is: $A + 4 \times \frac{5}{12} a^2 = a^2 + \frac{5}{3} a^2 = \frac{8}{3} a^2$ Given that the total surface area is 1536 cm²: $\frac{8}{3} a^2 = 1536$ Solving for $a^2$: $a^2 = \frac{1536 \times 3}{8} = 576 \quad \Rightarrow \quad a = \sqrt{576} = 24 \, \text{cm}$

Step 3: Solve for the Volume

Now, with $a = 24 \, \text{cm}$ and $l = \frac{5}{6} \times 24 = 20 \, \text{cm}$, we can use the Pythagorean theorem to find $h$: $h^2 + \left(\frac{24}{2}\right)^2 = 20^2$ $h^2 + 12^2 = 20^2$ $h^2 + 144 = 400 \quad \Rightarrow \quad h^2 = 256 \quad \Rightarrow \quad h = \sqrt{256} = 16 \, \text{cm}$

Finally, the volume of the pyramid is: $V = \frac{1}{3} \times 24^2 \times 16 = \frac{1}{3} \times 576 \times 16 = \frac{9216}{3} = 3072 \, \text{cm}^3$

Final Relations:

1. Base Area: $A = a^2 = 576 \, \text{cm}^2$.
2. Height: $h = 16 \, \text{cm}$.
3. Volume: $V = 3072 \, \text{cm}^3$.

Would you like to explore further details, or do you have any specific questions?

Additional Questions:

1. What formula links the slant height to the height of a pyramid?
2. How does the surface area of a square pyramid differ from other pyramids?
3. Can you derive the formula for the volume of any pyramid?
4. What are the units for volume and how do they relate to area and height?
5. What other shapes could have similar surface area calculations?

Tip: Always use the Pythagorean theorem to relate slant height and height in pyramid problems.