Given:
volume - 32 cubic units
height of the pyramid - 6 units
width of the base - 8 units
Find: height of the base (x)
Solution:
In order to find the height of the base, let's recall the formula of the volume of a triangular pyramid.
[tex]V=\frac{1}{3}(BaseArea\times Pyramid^{\prime}sHeight)[/tex]
To get the height of the base, let's solve for the Area of the Base first.
Let's plug into the formula the value of the volume and the pyramid's height to solve for the Base Area.
[tex]\begin{gathered} 32=\frac{1}{3}(x\times6) \\ Cross-multiply. \\ 96=6x \\ Divide\text{ }6\text{ }on\text{ }both\text{ }sides\text{ }of\text{ }the\text{ }equation. \\ \frac{96}{6}=\frac{6x}{6} \\ 16=x \end{gathered}[/tex]
Hence, the area of the base is 16 square units.
To get the height of the base, let's recall the formula for getting the area of the base triangle.
[tex]A_{triangle}=\frac{1}{2}(base\times height)[/tex]
Now, let's plug into the formula above the value of the area of the base and the given base width.
[tex]\begin{gathered} 16=\frac{1}{2}(8\times h) \\ Cross-multiply. \\ 32=8h \\ Divide\text{ }both\text{ }sides\text{ }of\text{ }the\text{ }equation\text{ }by\text{ }8. \\ \frac{32}{8}=\frac{8h}{8} \\ 4=h \end{gathered}[/tex]
Answer:
The height of the base (x) is 4 units.
