To calculate the volume of the given rectangular pyramid, we use the formula for the volume of a pyramid:

$V = \frac{1}{3} \cdot \text{Base Area} \cdot \text{Height}$

Step 1: Identify the dimensions

From the image:

• The base is a rectangle with dimensions $9 \, \text{yd}$ and $7 \, \text{yd}$.
• The slant height is $7 \, \text{yd}$, but we need the perpendicular height $h$ of the pyramid.

Step 2: Calculate the base area

The area of the base is: $\text{Base Area} = \text{Length} \cdot \text{Width} = 9 \cdot 7 = 63 \, \text{yd}^2$

Step 3: Find the perpendicular height

Using the slant height and the base's diagonal, we can calculate the height $h$. The diagonal of the base is given by: $\text{Diagonal of base} = \sqrt{9^2 + 7^2} = \sqrt{81 + 49} = \sqrt{130} \, \text{yd}$

The slant height forms a right triangle with the height $h$ and half the diagonal of the base. Thus: $\left(\frac{\text{Diagonal of base}}{2}\right)^2 + h^2 = \text{Slant height}^2$ $\left(\frac{\sqrt{130}}{2}\right)^2 + h^2 = 7^2$ $\frac{130}{4} + h^2 = 49$ $32.5 + h^2 = 49$ $h^2 = 16.5$ $h = \sqrt{16.5} \approx 4.06 \, \text{yd}$

Step 4: Calculate the volume

Now substitute the values into the volume formula: $V = \frac{1}{3} \cdot 63 \cdot 4.06$ $V \approx 85.38 \, \text{yd}^3$

Final Answer:

The volume of the pyramid is approximately: $\boxed{85.38 \, \text{yd}^3}$

Would you like further clarification or more examples? Here are five related questions to expand your understanding:

1. How does the slant height differ from the perpendicular height in a pyramid?
2. Can this formula be applied to pyramids with other base shapes (e.g., triangular bases)?
3. What role does the diagonal of the base play in determining the height of the pyramid?
4. How would the volume change if the dimensions of the base were doubled?
5. What is the relationship between the slant height, base diagonal, and perpendicular height in a pyramid?

Tip: Always sketch and label all dimensions when solving 3D geometry problems—it helps visualize relationships clearly!