Which best describes the relationship among the volumes of hemisphere X, cylinder Y, and cone Z?

A. The sum of the volumes of X and Y equals the volume of Z.

B. The sum of the volume of X and Z equals the volume of Y.

C. The difference of the volumes of Z and Y equals the volume of X.

D. The difference of the volumes of X and Z equals the volume of Y.

B. The sum of the volume of X and Z equals the volume of Y.

Step-by-step explanation:

To solve the problem, we calculate the volume of the three solids, and then we compare them.

The volume of X is the volume of a hemisphere of radius r, which is half the volume of a sphere of radius r, so:

The volume of Y is the volume of a cylinder of radius r and height

h = r

so it is given by the formula:

Finally, the volume of Z is the volume of a cone of radius r and height

h = r

So the volume is given by

So we see that the correct option is

B. The sum of the volume of X and Z equals the volume of Y.

In fact:

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

the figures below are made out of circles, semicircles, quarter … find the area and the perimeter of each figure and give your answers as a completely simplified exact value in terms of π (no approximations). remember that we'll also have to cut whatever areas we get in half, since we're only dealing

step-by-step explanation: