A solid has a base bounded by x^2_y^2=36. Find the volume of the solid if every plane section perpendicular to the diameter is an isosceles triangle whose base is on the circle and whose height is 4 units
since the area of a triangle is 1/2 base * height, if the base is 2y, then we need only double the volume of half the solid, due to symmetry.
v = 2∫[0,6] (1/2)(2√(36-x^2))(4) dx
= 8∫[0,6] √(36-x^2) dx
Now just plug and chug.
To find the volume of the solid, we need to determine the shape formed by the intersecting planes when they cut through the solid.
Given that every plane section perpendicular to the diameter is an isosceles triangle with a base on the circle and a height of 4 units, we can conclude that the solid is a cone.
The equation x^2 + y^2 = 36 represents a circle with a radius of 6 units centered at the origin.
Using the formula for the volume of a cone, V = (1/3)πr^2h, we can calculate the volume of the solid.
Since the circle is the base of the solid and we know that the height of each triangular slice is 4 units, we need to determine the radius of each triangular slice.
The radius is the distance from the center of the circle to the base of the triangle. In this case, the height of the triangle is perpendicular to the diameter of the circle. Since the height of each triangular slice is 4 units, the radius of each triangular slice is also 4 units.
Substituting the values into the formula, V = (1/3)π(4^2)(4), we can simplify the calculation as follows:
V = (1/3)π(16)(4)
V = (1/3)π(64)
V = (1/3)(64π)
V = 64π/3
Therefore, the volume of the solid is 64π/3 cubic units.
To find the volume of the solid, we need to first determine the shape of the solid.
Given that each plane section perpendicular to the diameter is an isosceles triangle with a base on the circle and a height of 4 units, we can conclude that the shape of the solid is a cone with a circular base.
To find the volume of a cone, we use the formula:
Volume = (1/3) * π * r^2 * h
where r is the radius of the base and h is the height of the cone.
In this case, the base of the cone is bounded by the equation x^2 + y^2 = 36. Since this is a circle centered at the origin, the radius is simply the square root of the constant term, which is 6.
The height of the cone is given as 4 units.
Now we can substitute the values into the formula to find the volume:
Volume = (1/3) * π * 6^2 * 4
= (1/3) * π * 36 * 4
= (12/3) * π * 36
= 4 * π * 36
= 144π
Therefore, the volume of the solid is 144π cubic units.