# Thread: Volume of a solid

1. ## Volume of a solid

The base of a certain solid is an equilateral triangle with altitude 9. Cross-sections perpendicular to the altitude are semicircles. Find the volume of the solid, using the formula
applied to the picture shown above (click for a better view), with the left vertex of the triangle at the origin and the given altitude along the -axis.
Note: You can get full credit for this problem by just entering the final answer (to the last question) correctly. The initial questions are meant as hints towards the final answer and also allow you the opportunity to get partial credit.
The lower limit of integration is =
The upper limit of integration is =
The diameter of the semicircular cross-section is the following function of :
=
Thus the volume of the solid is =

2. let the left vertex be at the origin ...

The lower limit of integration is $\displaystyle a = 0$
The upper limit of integration is $\displaystyle b = 9$

slicing the equilateral triangle in half horizontally, two 30-60-90 triangles are formed ... run is 9 , rise is $\displaystyle 3\sqrt{3}$

$\displaystyle f(x) = \frac{\sqrt{3}}{3} x$

$\displaystyle g(x) = -\frac{\sqrt{3}}{3} x$

$\displaystyle 2r = f(x) - g(x) = \frac{2\sqrt{3}}{3} x$

$\displaystyle r = \frac{\sqrt{3}}{3} x$

$\displaystyle r^2 = \frac{x^2}{3}$

$\displaystyle A = \frac{\pi r^2}{2} = \frac{\pi x^2}{6}$

all the grunt work is done ... you finish up.