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 formulaV=∫abA(x), with the left vertex of the triangle at the origin and the given altitude along the x-axis.

To find the volume of the solid using the given information, we can follow these steps:

### Step 1: Determine the equation of the base
The base of the solid is an equilateral triangle with an altitude of 9 units. Since the triangle is equilateral, the length of each side can be determined using the formula for the height (altitude) of an equilateral triangle:

\[
\text{Height} = \frac{\sqrt{3}}{2} \times \text{side length}
\]

Given that the height is 9, we can solve for the side length \( s \):

\[
9 = \frac{\sqrt{3}}{2} s
\]

\[
s = \frac{18}{\sqrt{3}} = 6\sqrt{3}
\]

Since the altitude lies along the \( x \)-axis and the left vertex of the triangle is at the origin, the vertices of the triangle are located at \( (0, 0) \), \( (9, 0) \), and \( \left(\frac{9}{2}, 6\sqrt{3}\right) \).

### Step 2: Determine the function for the base width
The width of the triangle at any height \( x \) from the base (at \( x = 0 \)) can be determined by the line equation. The length of the side perpendicular to the altitude at a distance \( x \) from the base is given by:

\[
\text{Width}(x) = 2 \times \text{length of the segment from the center to the edge}


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 V = ∫[a,b] A(x), with the left vertex of the triangle at the origin and the given altitude along the x-axis.

Discover how to calculate the volume of a solid with an equilateral triangle base and semicircular cross-sections. This problem involves geometry and calculus concepts, such as integral calculus and geometric properties of triangles, suitable for students in Grades 11-12.

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