The base of a solid is a circle of
radius = 4
Find the exact volume of this solid if the cross sections perpendicular to a given axis are equilateral right triangles.
I have the area of the triangle (1/2bh) to be equal to 2sqrt(12)
(1/2 * 4 * sqrt12)
I don't know what to do next. If someone knows, I'd be very thankful.
there are like triangles that have vertical bases. They run parallel to the y-axis. The triangles form 2 triangles each that are symmetric to the x-axis.
The equation of the circle is:
x^2 + y^2 = 16
Thanks!
answer are
1.rectangular prism
2.pyramid
3. BE and DF
4.first one
5.first one
6.rectangular prism
7.cube
8.2,280 cm^2
9.302 m^2
10.715 in^3
11.216 ft^3
12.151 in.^3
13.147 yd^3
14.10 cm
15.16 cm
16. 9 ft
17.810 cm^3
18. 201 m^2
19. 268^3
To find the volume of the solid with equilateral right triangle cross sections, we can use integration.
First, let's consider a cross section of the solid perpendicular to the y-axis. This cross section will be an equilateral right triangle with a base along the x-axis.
From the equation of the circle, x^2 + y^2 = 16, we can solve for y in terms of x:
y = sqrt(16 - x^2)
Since the base of the triangle lies along the x-axis, the length of the base will vary with x.
From the given information, we know that the area of the triangle is 2sqrt(12).
The area of an equilateral triangle is (side length)^2 * sqrt(3) / 4.
Given that the base of the triangle is x, we can express the side length of the equilateral triangle as 2x/sqrt(3).
Therefore, we have:
(2x/sqrt(3))^2 * sqrt(3) / 4 = 2sqrt(12)
Simplifying this equation gives us:
4x^2 / 3 = 2sqrt(12)
After simplifying further, we get:
2x^2 = 3sqrt(12)
Now let's find the limits of integration for x. The circle has a radius of 4, so the x-values will go from -4 to 4.
Therefore, the volume of the solid can be calculated as the integral of the cross-sectional area over the range of x:
V = ∫[from -4 to 4] (2x^2) dx
Integrating this expression will give us the volume of the solid:
V = [2/3 * x^3] from -4 to 4
Evaluating this expression gives us the final answer for the exact volume of the solid with equilateral right triangle cross sections:
V = (2/3 * 4^3) - (2/3 * (-4)^3) = (128/3) - (-128/3) = 256/3
To find the volume of the solid whose cross sections perpendicular to a given axis are equilateral right triangles, we need to set up an integral based on the equation of the circle and the area of the triangle.
Since the cross sections are equilateral right triangles, each cross section will have a height and base of equal length, denoted by h. Let's use the y-axis as the given axis, meaning the cross sections will be perpendicular to the y-axis.
First, let's find the limits of integration. Since the base of the solid is a circle with a radius of 4, we can consider the range of y-values that lie within the circle. By substituting x^2 + y^2 = 16 into the equation, we get x^2 + 16 - y^2 = 16. Simplifying gives us x^2 = y^2. Taking the square root of both sides and rearranging, we get x = ±y.
Thus, the limits of integration for y will be from -4 to 4.
Next, we need to express the height of each triangle in terms of y. Since the cross sections are equilateral triangles, the formula for the height of an equilateral triangle is given by h = √(3/4) * s, where s is the length of one side of the triangle.
In this case, since the triangles are right triangles, the length of one side will be equal to the y-coordinate at each point. Therefore, the height of each triangle will be h = √(3/4) * |y|.
Now, we can set up the integral to find the volume. The volume element for each cross section will be the area of the triangle times a small differential element dy. Thus, we integrate the area of the triangle multiplied by dy from the lower limit to the upper limit:
V = ∫[from -4 to 4] (1/2 * base * height) dy
= ∫[from -4 to 4] (1/2 * |y| * √(3/4) * |y|) dy
= √(3/4) * ∫[from -4 to 4] (1/2 * y^2) dy
= √(3/4) * (1/2) * ∫[from -4 to 4] y^2 dy
Evaluating the integral, we get:
V = √(3/4) * (1/2) * (1/3) * [y^3] [from -4 to 4]
= √(3/4) * (1/6) * [4^3 - (-4)^3]
= √(3/4) * (1/6) * (64 - (-64))
= √(3/4) * (1/6) * 128
= 16√(3)
Therefore, the exact volume of the solid is 16√(3) cubic units.
I hope this explanation helps! Let me know if you have any further questions.