Question

a cylinder has a radius of 4 cm and a height of 7 cm. which is closest to its volume? (π ≈ 3.14)
a. 450 cm³
b. 351.68 cm³
c. 280.79 cm³
d. 400 cm³

what does the term \cross - sectional area\ refer to in the context of cavalieri’s principle?
a. the total surface area of a solid
b. the area of a two - dimensional slice of a solid at a given height
c. the length of the base of a solid
d. the volume of a solid at a specific height

which of the following is a correct application of cavalieri’s principle?
a. verifying the volume of a tilted stack of circular disks
b. calculating the perimeter of a rectangle
c. measuring the height of a triangle
d. determining the surface area of a cylinder

cavalieri’s principle states that two solids have the same volume if they have the same ______ and their corresponding cross - sectional areas are equal at every height.
a. shape
b. height
c. density
d. surface area

Explanation:

Step1: Recall cylinder volume formula

The volume $V$ of a cylinder is $V = \pi r^2 h$, where $r$ is radius, $h$ is height.

Step2: Substitute given values

$r=4\ \text{cm}$, $h=7\ \text{cm}$, $\pi=3.14$
$V = 3.14 \times 4^2 \times 7$

Step3: Calculate step-by-step

First compute $4^2 = 16$, then $3.14 \times 16 = 50.24$, finally $50.24 \times 7 = 351.68$
---

Step1: Define cross-sectional area (Cavalieri's)

Cross-sectional area in this principle is the 2D slice area at a height.

Step2: Match to correct option

Select the option describing this definition.
---

Step1: Recall Cavalieri's Principle use

It applies to volume of solids with matching cross-sections/height.

Step2: Identify valid application

Tilted disk stack (a cylinder variant) uses this for volume.
---

Step1: Recall Cavalieri's Principle statement

Two solids have equal volume if same height and equal cross-sections at all heights.

Step2: Fill in the missing term

Identify the required matching property.

Answer:

1. b. $351.68\ \text{cm}^3$
2. b. The area of a two-dimensional slice of a solid at a given height
3. a. Verifying the volume of a tilted stack of circular disks
4. b. Height