To find the volume of the oblique pyramid with a square base, an edge length of 2 cm, and an angle BAC of [tex]\(45^{\circ}\)[/tex], we need to follow these steps:
1. Understand the Geometry:
- The base of the pyramid is a square with each side measuring 2 cm.
- The angle BAC, which is the angle between the apex (vertex A) of the pyramid and one side of the base (side BC), is [tex]\(45^{\circ}\)[/tex].
2. Calculate the Height:
- The height of the pyramid can be inferred using trigonometry. Since angle BAC is [tex]\(45^{\circ}\)[/tex], tan(45°) = 1.
- We can use this angle to determine the height (h) from the apex to the center of the base. If we drop a perpendicular from the apex to the center of the base, this creates a right triangle where:
[tex]\[ \tan(45^{\circ}) = \frac{\text{height}}{\text{half of base edge}} \][/tex]
Since [tex]\(\tan(45^{\circ}) = 1\)[/tex]:
[tex]\[ 1 = \frac{h}{1 cm} \][/tex]
Therefore,
[tex]\[ h = 1 \][/tex]
3. Calculate the Area of the Base:
- The base is a square with each side of 2 cm:
[tex]\[ \text{Area of the base} = \text{side}^2 = 2^2 = 4 \, \text{cm}^2 \][/tex]
4. Calculate the Volume:
- The volume (V) of a pyramid is given by:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
- Substituting the known values:
[tex]\[ V = \frac{1}{3} \times 4 \times 1 = \frac{4}{3} \][/tex]
- Therefore,
[tex]\[ V \approx 1.33 \, \text{cm}^3 \][/tex]
Given the options provided ([tex]\(2.4 \, \text{cm}^3\)[/tex], [tex]\(3.6 \, \text{cm}^3\)[/tex], [tex]\(4.8 \, \text{cm}^3\)[/tex], [tex]\(7.2 \, \text{cm}^3\)[/tex]), the correct approximate volume of the pyramid is [tex]\(1.33 \, \text{cm}^3\)[/tex].
Since none of the given options match the exact volume of [tex]\( 1.33 \, \text{cm}^3 \)[/tex], it appears there might be a mistake in the provided options or a rounding issue. The correct volume based on our calculations is [tex]\( \boxed{1.33 \, \text{cm}^3} \)[/tex].
