Answer:
20. 170 cm³
21. The ice cream sandwich (rectangular prism) has the greatest volume.
Step-by-step explanation:
20. You are asked to find the difference between the volumes of three spheres and the volume of the cylinder containing them. It is appropriate to make use of the formulas for the volume of a sphere and the volume of a cylinder.
<u>Sphere</u>
The volume is given by ...
V = (4/3)πr³
You have 3 spheres, each with a radius of 3 cm, so their total volume is ...
3V = 3·(4/3)·π·(3 cm)³ = 108π cm³
<u>Cylinder</u>
The volume is given by ...
V = πr²·h
You have a radius of 3 cm and a height of 18 cm, so the volume of the cylinder is ...
V = π·(3 cm)²·(18 cm) = 162π cm³
<u>Empty Space</u>
The volume of the space in the cylinder not occupied by the balls will be ...
space = (162π cm³) - (108π cm³) = 54π cm³ ≈ 169.64 cm³ ≈ 170 cm³
The space not occupied by balls is 170 cm³.
__
21. The formula for the volume of a cone with a certain radius and height is ...
V = (1/3)πr²·h
If we let r = d/2, where d is the diameter, then the volume is ...
V = (1/3)π(d/2)²·h = π/12·d²·h ≈ 0.262·d²·h
The formula for the volume of a cylinder with a certain radius and height is ...
V = πr²·h
Again, using r = d/2, the volume is ...
V = π(d/2)²·h = (π/4)·d²·h ≈ 0.785·d²·h
The formula for the volume of a rectangular prism with a side length of d and a height of h is ...
V = d²·h
That is, the volume of the cone (not counting the ice cream on top) is about 0.262 that of the prism, and the volume of the cylinder is about 0.785 that of the prism. Obviously, the prism shape contains the greatest volume.
___
The conclusion doesn't change if we add the volume of a hemisphere of ice cream on top of the cone. That volume is ...
hemisphere volume = (2/3)πr³ = (2/3)π(d/2)³ = π/12·d³ = π/12·d²·d
Now, the volume of the cone and hemisphere together will be ...
topped cone volume = π/12·d²·h + π/12·d²·d = (π/12)·d²·(h+d)
That is, we have multiplied our previous assessment of the volume of the ice cream cone by (h+d)/h = (6+4)/6 = 5/3. This makes the multiplier of d²·h be about π/12·(5/3) ≈ 0.436, still less than half the volume of the ice cream bar.
___
Note that we have chosen to look at the relative sizes of the cone, cylinder and prism in a generic way. You can fill in the numbers and compare volumes in cubic inches. The conclusion will be the same. We hope this method helps develop some intuition about the relative sizes of cones, cylinders, and prisms that have the same dimensions.
