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?
36 sqrt 3
36
18 sqrt 3
18
The answer isn't 18 sqrt 3 for sure.
Consider the triangle at distance x. Its base is 2y, and its height is y√3. So, adding up the volumes of all those triangles of thickness dx, we have (applying symmetry)
v = 2∫[0,3] 1/2 * 2y * y√3 dx
= 2√3 ∫[0,3] y^2 dx
= 2√3 ∫[0,3] (9-x^2) dx
= 36√3
Well, aren't we "tri"-ing to figure this out? Let me break it down for you. Since the cross sections perpendicular to the x-axis are equilateral triangles, we know that each side of these triangles is determined by the y-coordinate.
The given equation, x^2 + y^2 = 9, represents a circle with radius 3. Since we are only interested in the portion of the circle that lies in the positive y-axis region, we can take the upper half of the circle from y = 0 to y = 3.
The side length of the equilateral triangle is equal to 2y, and ranges from 0 to 3. So, the area of each cross section is A = (sqrt(3)/4)(2y)^2 = (3sqrt(3)/2)y^2.
To find the volume of the solid, we need to integrate the area function from y = 0 to y = 3. By integrating A = (3sqrt(3)/2)y^2 with respect to y, we get V = (sqrt(3)/8)y^3 | from 0 to 3.
Evaluating this integral, we find V = (sqrt(3)/8)(3)^3 - (sqrt(3)/8)(0)^3 = (sqrt(3)/8)(27) = 9sqrt(3)/8.
So, the volume of the solid is 9sqrt(3)/8 cubic units. Hence, the correct answer is 9sqrt(3)/8.
Hope that clears things up and doesn't give you any "triangle-ger."
To find the volume of the solid, we need to determine the area of each cross section and then integrate over the entire range of x-values.
Since the cross sections are equilateral triangles, we need to find the side length of these triangles as a function of x.
The equation of the given circle is x^2 + y^2 = 9. By rearranging this equation, we can solve for y:
y = sqrt(9 - x^2) or y = -sqrt(9 - x^2)
To find the side length of the equilateral triangle, we need to know the length of the perpendicular line segment from the center of the circle to the edge of the triangle. Let's call this length "h."
Consider an arbitrary point on the circle (x, y). The distance from this point to the x-axis is y. We can draw a right triangle with y as the hypotenuse and h as the height of the triangle. The base of this right triangle will be half of the side length of the equilateral triangle since the triangle is symmetric about the y-axis.
Using the Pythagorean theorem, we can find h in terms of x:
h = sqrt(y^2 - (side length/2)^2)
h = sqrt(y^2 - (y/sqrt(3))^2)
h = sqrt(y^2 - y^2/3)
h = sqrt(2y^2/3)
h = sqrt(2/3) y
Now, we can find the area of the equilateral triangle as a function of x:
Area = (base * height)/2
Area = (side length^2 * sqrt(3)/4)/2
Area = side length^2 * sqrt(3)/8
Since the base of the solid is the circle x^2 + y^2 = 9, we need to find the limits of integration for x. This can be done by finding the x-coordinates of the points where the circle intersects the x-axis:
x^2 + y^2 = 9
x^2 + 0^2 = 9
x^2 = 9
x = -3, 3
The volume V of the solid is given by the integral of the area of the cross section from -3 to 3:
V = ∫[from -3 to 3] (side length(x)^2 * sqrt(3)/8) dx
To find the side length as a function of x, we substitute y = sqrt(9 - x^2) into our equation for h:
h = sqrt(2/3) * sqrt(9 - x^2)
h = sqrt(18 - 2x^2/3)
Substituting this expression for h into the equation for the area of the cross section, we have:
Area = (side length(x)^2 * sqrt(3)/8)
Area = (sqrt(18 - 2x^2/3))^2 * sqrt(3)/8
Area = (3(18 - 2x^2/3))/24 * sqrt(3)
Area = (1/8)(18 - 2x^2/3) * sqrt(3)
Area = (3sqrt(3) - x^2sqrt(3)/12)
Now, we can integrate the area function from -3 to 3 to find the volume:
V = ∫[from -3 to 3] (3sqrt(3) - x^2sqrt(3)/12) dx
Evaluating this integral, we have:
V = [3sqrt(3)x - (x^3/12)sqrt(3)] | [from -3 to 3]
V = [3sqrt(3) * 3 - (3^3/12)sqrt(3)] - [3sqrt(3) * -3 - ((-3)^3/12)sqrt(3)]
V = [9sqrt(3) - (27/12)sqrt(3)] - [-9sqrt(3) - (27/12)sqrt(3)]
V = [9sqrt(3) - 9sqrt(3)/4] - [-9sqrt(3) - 9sqrt(3)/4]
V = 9sqrt(3) - 9sqrt(3)/4 + 9sqrt(3) + 9sqrt(3)/4
V = 36sqrt(3)
Therefore, the volume of the solid is 36sqrt(3) cubic units.
To find the volume of the solid, we need to integrate the area of each cross section perpendicular to the x-axis.
First, let's analyze the geometric properties of the equilateral triangles. The cross sections perpendicular to the x-axis are equilateral triangles, which means all angles are 60 degrees (or π/3 radians), and all sides have equal length.
Given that the base of the solid is the circle x² + y² = 9, we can see that it has a radius of 3 units. Since the triangle is equilateral and the circle is centered at the origin, the height of each triangle will be the distance from the x-axis to the topmost or bottommost vertex of the triangle.
To find the height of the equilateral triangle, we can draw a line from the center of the circle to the top vertex. This line will be perpendicular to the base of the triangle, forming a right triangle. The base of this right triangle will be half the length of one side of the triangle, which is (√3)/2 times the length of the side.
Using the Pythagorean theorem, we can calculate the height of the equilateral triangle:
height² + base² = side²
height² + ((√3)/2 * side)² = side²
height² + (3/4) * side² = side²
height² = side² - (3/4) * side²
height² = (1 - (3/4)) * side²
height² = (1/4) * side²
height = (1/2) * side
Now, we can calculate the area of each equilateral triangle. The area of an equilateral triangle is given by the formula:
Area = (sqrt(3) / 4) * side²
Since the side length of our equilateral triangle is equal to the diameter of the circle, which is 3 units, we can substitute this value into the formula:
Area = (sqrt(3) / 4) * (3)² = (sqrt(3) / 4) * 9 = (3sqrt(3)) / 4
To find the volume of the solid, we need to sum up the areas of all the equilateral triangles. Since the solid extends from x = -3 to x = 3, we need to integrate the area function from -3 to 3:
Volume = ∫[-3, 3] [(3sqrt(3)) / 4] dx
Integrating, we get:
Volume = [(3sqrt(3)) / 4] * x ∣[-3, 3]
Volume = [(3sqrt(3)) / 4] * (3 - (-3))
Volume = (9sqrt(3)) / 4 * 6
Volume = (27sqrt(3)) / 2
Therefore, the volume of the solid is (27sqrt(3)) / 2 cubic units. None of the provided options match this answer, so there may be an error in the answer choices given.