The base of a solid is the circle x2 + y2 = 9. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?
To find the volume of the solid, we need to integrate the area of the equilateral triangles along the x-axis.
First, let's determine the length of the base of the equilateral triangle formed by the cross section. Since the cross section is perpendicular to the x-axis, the base of the equilateral triangle will be twice the radius of the circle.
The equation of the circle is x^2 + y^2 = 9. Since the circle is centered at the origin (0,0), the radius is √9 = 3. Hence, the base of the equilateral triangle is 2 * 3 = 6.
Next, we need to determine the height of the equilateral triangle. By connecting the center of the circle to any point on its circumference, we form a radius that is also the height of the equilateral triangle. Using the Pythagorean theorem, we can find the height.
Let's take a point on the circumference of the circle, (x,y). The distance from the origin to this point is the radius, which is 3. The x-coordinate of this point will be the base of the triangle, so it is 6. The y-coordinate can be found using the equation of the circle: x^2 + y^2 = 9. Substituting x=6, we have 6^2 + y^2 = 9, giving us y = ±√3.
Hence, the height of the equilateral triangle is √3.
Since the volume of a prism is given by V = area of base * height, and the cross sections are equilateral triangles, the area of the base is (1/2) * base * height. Substituting the values we found, V = (1/2) * 6 * √3 = 3√3.
Therefore, the volume of the solid is 3√3 cubic units.
To determine the volume of the solid, we need to find the area of each cross section perpendicular to the x-axis and then integrate it over the range of x-values that intersect the base circle.
First, let's visualize the situation: the base of the solid is a circle with radius 3 centered at the origin. The cross sections perpendicular to the x-axis are equilateral triangles.
To find the area of an individual equilateral triangle, we need to know the length of its side. Let's determine the side length by considering a cross section at a specific x-coordinate.
For any given x-coordinate, the corresponding y-coordinate on the circle can be found by solving the equation of the circle: x^2 + y^2 = 9. Rearranging this equation, we have: y^2 = 9 - x^2. Taking the square root of both sides, we get: y = ±√(9 - x^2).
Since the cross section is perpendicular to the x-axis, the side length of the equilateral triangle is equal to twice the y-coordinate corresponding to a given x-coordinate (because the triangle extends above and below the x-axis). So, the side length of the equilateral triangle at x is given by: s = 2 * √(9 - x^2).
Now that we have the side length of the equilateral triangle as a function of x, let's find the area of the equilateral triangle. The formula for the area of an equilateral triangle is: A = √3/4 * s^2, where s is the side length. Plugging in the expression for s, we have: A = √3/4 * (2 * √(9 - x^2))^2.
Simplifying, we have: A = √3/4 * 4 * (9 - x^2) = √3(9 - x^2).
To find the volume of the solid, we integrate the area function over the range of x-values that intersect the base circle. Since the base circle has a radius of 3, the x-values range from -3 to 3.
So, the volume of the solid can be calculated as follows:
V = ∫[from -3 to 3] √3(9 - x^2) dx
Integrating this expression, we get:
V = √3 * ∫[from -3 to 3] (9 - x^2) dx
V = √3 * [9x - (x^3)/3] | from -3 to 3
Evaluating this expression, we have:
V = √3 * [(9 * 3 - (3^3)/3) - (9 * -3 - (-3^3)/3)]
V = √3 * [27 - 9 - (-27 + 9)]
V = √3 * (54)
V = 54√3 cubic units
Therefore, the volume of the solid is 54√3 cubic units.
each triangle has a base 2y. So, since the area of an equilateral triangle of side s is √3/4 s^2, our triangles have area
√3/4 (9-x^2)
Now all we have to do is sum up all the triangular plates of thickness dx, from -3<=x<=3. Or, because of symmetry, twice the volume for 0<=x<=3.
v = 2∫[0,3] √3/4 (9-x^2) dx
and that's a cinch, right?