Find the volume of the solid whose base is the region bounded by the x axis, the curve y=(x^2+9)^(1/2), x=1,x=3 and which has the property that each cross section perpendicular to the x axis is equilateral triangle.

To find the volume of the solid, we can use integration. Since each cross section perpendicular to the x-axis is an equilateral triangle, we need to find the area of the equilateral triangle as a function of x.

Let's consider a small strip of thickness dx at a specific x-value. The length of this strip is given by the difference between the two y-values at x and x + dx, which is (x^2 + 9)^(1/2) - (x + dx)^2 + 9)^(1/2). The height of the equilateral triangle is equal to the distance between the x-axis and the y-value at x, which is (x^2 + 9)^(1/2).

Therefore, the area of the strip can be calculated as (1/2) * [(x^2 + 9)^(1/2)] * [(x^2 + 9)^(1/2) - (x + dx)^2 + 9)^(1/2)]. To get the volume, we need to integrate this area over the given interval.

The integral to calculate the volume is V = ∫[(1/2) * [(x^2 + 9)^(1/2)] * [(x^2 + 9)^(1/2) - (x + dx)^2 + 9)^(1/2)]] dx, with the limits of integration from x = 1 to x = 3.

Simplifying and integrating this expression will give us the volume of the solid.