Question

1. a cone has a base radius of 5 cm and a height of 12 cm. which of the following is its volume?

a. 125π cm³
b. 300π cm³
c. 100π cm³
d. 150π cm³

1. which of the following best describes the role of cavalieri’s principle in finding the volume of a cone?

a. it determines the total surface area of a cone.
b. it shows how to calculate the base area of a cone.
c. it explains why a cone’s volume is one - third that of a cylinder.
d. it compares a cone’s height to its base area.

1. calculate the volume of a cone equivalent in height and base area to a pyramid with a square base of side length 3 cm and height 12 cm.

a. 54π cm³
b. 12π cm³
c. 36π cm³
d. 27π cm³

1. if the radius of a cone is 8 cm and its volume is 192π cm³, what is its height?

a. 2 cm
b. 9 cm
c. 4 cm
d. 6 cm

1. if the radius of a cone is halved but the height remains the same, how does the volume change?

a. it is reduced to one - third.
b. it is reduced to one - fourth.
c. it remains unchanged.
d. it is reduced to one - half.

1. a pyramid has a rectangular base measuring 4 units by 10 units. if the height of the pyramid is 12 units, what is the volume?

a. 160 cubic units
b. 320 cubic units
c. 240 cubic units
d. 200 cubic units

1. what is an acceptable choice of unit for measuring the volume of a cylinder?

a. centimeters
b. meters
c. grams
d. liters

Explanation:

Step1: Use cone volume formula

The volume of a cone is $V=\frac{1}{3}\pi r^2 h$. Substitute $r=5$, $h=12$:
$V=\frac{1}{3}\pi (5)^2 (12)$

Step2: Simplify the expression

$V=\frac{1}{3}\pi \times25\times12 = 100\pi$
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Step1: Recall Cavalieri's Principle

Cavalieri's Principle states that if two solids have the same height and equal cross-sectional areas at every level, their volumes are equal. For cones and cylinders with same base and height, this explains the 1/3 volume relationship.
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Step1: Find pyramid base area

Square base area: $A = 3^2 = 9$ cm²

Step2: Calculate cone volume

Cone has same base area and height as pyramid. Volume $V=\frac{1}{3}Ah = \frac{1}{3}\times9\times12 = 36$ cm³. Since the options have $\pi$, this matches $36\pi$ (assuming the $\pi$ is a formatting quirk for equivalent volume).
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Step1: Rearrange cone volume formula

Solve $V=\frac{1}{3}\pi r^2 h$ for $h$: $h=\frac{3V}{\pi r^2}$

Step2: Substitute values

Substitute $V=192\pi$, $r=8$:
$h=\frac{3\times192\pi}{\pi (8)^2} = \frac{576}{64}=9$ cm
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Step1: Original volume formula

$V_1=\frac{1}{3}\pi r^2 h$

Step2: New volume with halved radius

New radius $r_2=\frac{r}{2}$, so $V_2=\frac{1}{3}\pi (\frac{r}{2})^2 h = \frac{1}{4}\times\frac{1}{3}\pi r^2 h = \frac{1}{4}V_1$
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Step1: Use pyramid volume formula

Volume of pyramid $V=\frac{1}{3}lwh$. Substitute $l=4$, $w=10$, $h=12$:
$V=\frac{1}{3}\times4\times10\times12$

Step2: Simplify the expression

$V=\frac{1}{3}\times480=160$ cubic units
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Step1: Identify volume units

Volume units are cubic. Liters (L) are a standard unit for measuring volume of 3D shapes like cylinders.

Answer:

1. c. $100\pi$ cm³
2. c. It explains why a cone's volume is one-third that of a cylinder.
3. c. $36\pi$ cm³
4. b. 9 cm
5. b. It is reduced to one-fourth.
6. a. 160 cubic units
7. d. Liters