Question

each row of this table lists the dimensions of a different cone.

1. complete the table.

$v=\frac{1}{3}\pi r^{2}$

1. discuss: how are determining the unknown dimensions of a cone and cylinder alike and different?

Explanation:

For Question 4 (Completing the Table)

Row 1

Step1: Calculate base area

Base area of a circle: $\pi r^2$
$\pi (4)^2 = 16\pi$

Step2: Calculate cylinder volume

Cylinder volume: $\pi r^2 h$
$16\pi \times 3 = 48\pi$

Step3: Calculate cone volume

Cone volume: $\frac{1}{3}\pi r^2 h$
$\frac{1}{3} \times 48\pi = 16\pi$

Row 2

Step1: Find radius from base area

Base area $\pi r^2 = 36\pi$, so $r^2=36$, $r=6$

Step2: Calculate diameter

Diameter = $2r$
$2\times6=12$

Step3: Calculate cylinder volume

Cylinder volume: $\pi r^2 h$
$36\pi \times \frac{1}{4} = 9\pi$

Step4: Calculate cone volume

Cone volume: $\frac{1}{3}\pi r^2 h$
$\frac{1}{3} \times 9\pi = 3\pi$

Row 3

Step1: Find radius from diameter

Radius $r=\frac{\text{Diameter}}{2}$
$\frac{20}{2}=10$

Step2: Calculate base area

Base area: $\pi r^2$
$\pi (10)^2=100\pi$

Step3: Find height from cone volume

Cone volume $V=\frac{1}{3}\pi r^2 h$, solve for $h$:
$h=\frac{3V}{\pi r^2} = \frac{3\times200\pi}{100\pi}=6$

Step4: Calculate cylinder volume

Cylinder volume: $\pi r^2 h$
$100\pi \times6=600\pi$

Row 4

Step1: Find base area from cone volume

Cone volume $V=\frac{1}{3}Bh$, solve for $B$ (base area):
$B=\frac{3V}{h}=\frac{3\times64\pi}{12}=16\pi$

Step2: Find radius from base area

$\pi r^2=16\pi$, so $r^2=16$, $r=4$

Step3: Calculate diameter

Diameter = $2r$
$2\times4=8$

Step4: Calculate cylinder volume

Cylinder volume: $Bh$
$16\pi \times12=192\pi$

• Alike: Both cones and cylinders use the same base area formula ($\pi r^2$) for circular bases, and you rearrange volume formulas to solve for unknown dimensions (radius, height, base area). Both rely on the relationship between linear dimensions (radius/diameter) and area/volume.
• Different: The volume formulas differ: cylinder volume is $V=\pi r^2 h$, while cone volume is $V=\frac{1}{3}\pi r^2 h$, so solving for unknowns requires accounting for the $\frac{1}{3}$ factor for cones. When given volume, cones will have a different height/radius relationship than a cylinder with the same base and volume.

Answer:

Diameter (units)	Radius (units)	Base Area (sq. units)	Height (units)	Cylinder Volume (cu. units)	Cone Volume (cu. units)
12	6	$36\pi$	$\frac{1}{4}$	$9\pi$	$3\pi$
20	10	$100\pi$	6	$600\pi$	$200\pi$
8	4	$16\pi$	12	$192\pi$	$64\pi$

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For Question 5