Question

1. a solid cylinder is cut vertically into two equal parts. how does the volume of one part compare to the original?

a. it is one - third.
b. it is one - fourth.
c. it is one - half.
d. it remains the same.

1. if a cone and a cylinder have the same height and base radius, which statement is true about their volumes?

a. the cone’s volume is the same as the cylinder.
b. the cone’s volume is one - third that of the cylinder.
c. the cone’s volume is half that of the cylinder.
d. the cone’s volume is twice that of the cylinder.

1. according to cavalieri’s principle, when comparing two solids of equal height and cross - sectional area, what can be concluded about their volumes?

a. the solid with the smaller base area has a greater volume.
b. their volumes are different.
c. their volumes are equal.
d. the solid with the larger surface area has a greater volume.

1. if a cylinder has a radius of 3 cm and a height of 8 cm, what is its volume?

a. 72π cm³
b. 48π cm³
c. 48 cm³
d. 72 cm³

1. if the height of a cone is 10 cm and the radius is 5 cm, what is the area of the base?

a. 25π square cm
b. 10π square cm
c. 50π square cm
d. 100π square cm

1. what is the primary purpose of calculating the volume of a pyramid?

a. to estimate the surface area
b. to determine the number of edges
c. to find its weight
d. to determine the amount of space the pyramid occupies

Explanation:

Step1: Analyze Q22 (Cylinder cut into 2)

Original cylinder volume: $V_{original} = \pi r^2 h$. Cutting vertically into 2 equal parts gives each part volume: $\frac{1}{2} \pi r^2 h$.

Step2: Analyze Q23 (Cone vs Cylinder volume)

Cylinder volume: $V_{cyl} = \pi r^2 h$. Cone volume: $V_{cone} = \frac{1}{3} \pi r^2 h$.

Step3: Analyze Q24 (Cavalieri's Principle)

By definition, solids with equal height and equal cross-sectional area at every height have equal volume.

Step4: Calculate Q25 (Cylinder volume)

Use $V = \pi r^2 h$. Substitute $r=3$, $h=5$: $V = \pi (3)^2 (5) = 45\pi$.

Step5: Calculate Q26 (Cone base area)

Base is a circle, area: $A = \pi r^2$. Substitute $r=5$: $A = \pi (5)^2 = 25\pi$.

Step6: Analyze Q27 (Volume purpose)

Volume measures the 3D space an object occupies.

Answer:

1. c. It is one-half
2. b. The cone's volume is one-third that of the cylinder
3. c. Their volumes are equal
4. c. $45\pi\ \text{cm}^3$
5. a. $25\pi$ square cm
6. d. To determine the amount of space the pyramid occupies