To find the height $h$ of the oblique pyramid, we need to use the formula for the volume of a pyramid. The volume $V$ of a pyramid is given by:

$V = \frac{1}{3} \times A_{\text{base}} \times h$

where:

• $V$ is the volume of the pyramid.
• $A_{\text{base}}$ is the area of the base of the pyramid.
• $h$ is the height of the pyramid.

From the problem:

• The volume $V$ is $13 \text{ cm}^3$.

The base of the pyramid is a triangle with sides of $4 \text{ cm}$, $3 \text{ cm}$, and $9 \text{ cm}$. However, these sides cannot form a triangle since the sum of the lengths of any two sides of a triangle must be greater than the length of the third side (triangle inequality). Since $4 + 3 = 7$ and $7 < 9$, this combination cannot make a triangle.

Assuming the provided sides are intended to form a valid triangle, let's reconsider the dimensions. If we assume the triangle sides were intended to be $4 \text{ cm}$ and $3 \text{ cm}$, but not the third side $9 \text{ cm}$, we might be given other base dimensions. If these were incorrect, let's use other base dimensions provided or assumed. However, we must move forward with the calculation using different assumptions to check for correctness.

Since it is important to use a valid triangle for the base, let us make a necessary assumption here and use a valid triangle.

1. Triangle Area Calculation Assumption: Let us assume we have two dimensions correctly, and the third dimension was incorrectly listed. Let's assume the triangle's base is 4 cm, and the height to this base is 3 cm.

2. Calculate the Area of Triangle Base:

   • Using base = 4 cm, height = 3 cm, the area of a triangle:

$A_{\text{base}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 3 = 6 \text{ cm}^2$

3. Plug in the Pyramid Volume Formula: Use the volume and the area:

$13 = \frac{1}{3} \times 6 \times h$

4. Solve for $h$:

$13 = 2h$

$h = \frac{13}{2} = 6.5 \text{ cm}$

Thus, the height $h$ of the pyramid is $\mathbf{6.5 \text{ cm}}$.

Would you like to revisit any part of the problem or have further questions?

Related Questions:

1. What is the triangle inequality theorem?
2. How do you calculate the area of a triangle with three given side lengths?
3. What are the properties of an oblique pyramid?
4. How do you determine the height of a pyramid if the volume and base area are known?
5. What are the differences between oblique and regular pyramids?

Tip:

Always check the validity of the triangle with the given side lengths before proceeding with calculations to ensure the dimensions form a valid shape.