A rectangular prism with a square base is cut from the center of a triangular prism. Each side of the square base measures 1.7 u
nits.
What is the volume of the shaded solid? Round to the nearest tenth of a cubic unit.
2 answers:
Let
be the volume of the solid S. We can write:
![V_{shaded} = V_{triangular} - V_{rectangular}](https://tex.z-dn.net/?f=V_%7Bshaded%7D%20%3D%20V_%7Btriangular%7D%20-%20V_%7Brectangular%7D)
Now, we just need to calculate the volume of the original prism (Vtriangular) and the volume that was removed (Vrectangular). So:
![\bullet ~V_{triangular} = S_{\triangle basis}\times h\\\\ V_{triangular} = \dfrac{5\times5.7}{2}\times4\\\\ V_{triangular} = 57~u^3\\\\\\ \bullet ~V_{rectangular} = S_{\square basis}\times h\\\\ V_{rectangular} = 1.7^2\times4\\\\ V_{rectangular} = 11.56~u^3](https://tex.z-dn.net/?f=%5Cbullet%20~V_%7Btriangular%7D%20%3D%20S_%7B%5Ctriangle%20basis%7D%5Ctimes%20h%5C%5C%5C%5C%0AV_%7Btriangular%7D%20%3D%20%5Cdfrac%7B5%5Ctimes5.7%7D%7B2%7D%5Ctimes4%5C%5C%5C%5C%0AV_%7Btriangular%7D%20%3D%2057~u%5E3%5C%5C%5C%5C%5C%5C%0A%5Cbullet%20~V_%7Brectangular%7D%20%3D%20S_%7B%5Csquare%20basis%7D%5Ctimes%20h%5C%5C%5C%5C%0AV_%7Brectangular%7D%20%3D%201.7%5E2%5Ctimes4%5C%5C%5C%5C%0AV_%7Brectangular%7D%20%3D%2011.56~u%5E3)
Then,
![V_{shaded} = V_{triangular} - V_{rectangular}\\\\ V_{shaded} = 57 - 11.56\\\\ V_{shaded} = 45.44~u^3\\\\ \boxed{V_{shaded} \approx 45.4~u^3}](https://tex.z-dn.net/?f=V_%7Bshaded%7D%20%3D%20V_%7Btriangular%7D%20-%20V_%7Brectangular%7D%5C%5C%5C%5C%0AV_%7Bshaded%7D%20%3D%2057%20-%2011.56%5C%5C%5C%5C%0AV_%7Bshaded%7D%20%3D%2045.44~u%5E3%5C%5C%5C%5C%0A%5Cboxed%7BV_%7Bshaded%7D%20%5Capprox%2045.4~u%5E3%7D)
Thus, the answer is 45.4 u³.
Answer:
the answer is 45.4 u³.
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