Question 2 (matching worth 10 points) (08.05 lc) match    term    definition triangle                  a) perpendicular cross section of a cylinder rectangle              b) perpendicular cross section of a pyramid circle                    c) shape created when a right triangle is rotated about the y-axis cylinder                d) parallel cross section of a sphere

Part A) Rectangle

Part B) Triangle

Part C) Cone

Part D) Circle

Step-by-step explanation:

Part A) Perpendicular cross section of a cylinder

The perpendicular cross section of a cylinder is a rectangle

The base of the rectangle is equal top the diameter of the circular base of cylinder and the height of the rectangle is equal to the height of the cylinder

Part B) Perpendicular cross section of a pyramid

Cross sections perpendicular to the base and through the vertex of a pyramid will be triangles

Part C) Shape created when a right triangle is rotated about the y-axis

When a right triangle is rotated about the y-axis, then it forms a cone.

The vertical leg of the triangle becomes the height of the cone.

The horizontal leg of the triangle becomes the radius of the cone.

The hypotenuse of the triangle becomes the slant height

Part D) Parallel cross section of a sphere

The parallel cross section of a sphere is a circle

All cross sections of a sphere are circles

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.