To find the volume of the composite figure, we need to calculate the volume of the rectangular prism and the volume of the two triangular prisms separately, and then add them together.
-
Volume of the Rectangular Prism: The volume \( V \) of a rectangular prism can be calculated using the formula: \[ V = \text{length} \times \text{width} \times \text{height} \] Given:
- Length = 14 cm
- Width = 5 cm
- Height = 4 cm
Plugging in the values: \[ V = 14 , \text{cm} \times 5 , \text{cm} \times 4 , \text{cm} = 280 , \text{cm}^3 \]
-
Volume of One Triangular Prism: The volume \( V \) of a triangular prism can be calculated using the formula: \[ V = \text{Base Area} \times \text{Height} \] First, we find the base area of the triangular prism. The base is a triangle and can be calculated as: \[ \text{Base Area} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base of the triangular prism is equal to the width of the rectangular prism, which is 5 cm, and the height from the apex to the base is given as 6 cm. Therefore, \[ \text{Base Area} = \frac{1}{2} \times 5 , \text{cm} \times 6 , \text{cm} = 15 , \text{cm}^2 \] Now, the height of the triangular prism (the length of the rectangular prism to which it is attached) is 14 cm.
Now we can find the volume of one triangular prism: \[ V = \text{Base Area} \times \text{height} = 15 , \text{cm}^2 \times 14 , \text{cm} = 210 , \text{cm}^3 \]
-
Volume of Two Triangular Prisms: Since there are two identical triangular prisms: \[ \text{Total volume of two triangular prisms} = 2 \times 210 , \text{cm}^3 = 420 , \text{cm}^3 \]
-
Total Volume of the Composite Figure: Now, we add the volumes of the rectangular prism and the two triangular prisms: \[ \text{Total Volume} = \text{Volume of Rectangular Prism} + \text{Volume of Two Triangular Prisms} = 280 , \text{cm}^3 + 420 , \text{cm}^3 = 700 , \text{cm}^3 \]
It appears my calculation of the individual component volumes was consistent, but I missed providing clarification on my final integration mistake.
After re-checking the individual computations:
- Rectangular Prism Volume = 280 cm³
- Combined Triangular Prisms Volume = 420 cm³
- The total of such reconciles to 700 cm³ and no options suit this inquired inquiry set.
Please clarify if original assumption figures and triangle metrics require revision. I apologize for any confusion or lapse on exhaustive rounding dataset counts.