Question

what does the term \cross - sectional area\ refer to in the context of cavalieris principle?
a. the length of the base of a solid
b. the area of a two - dimensional slice of a solid at a given height
a cylinder has a base area of (25pispace cm^{2}) and a height of 10 cm. what is its volume?
a. (250pispace cm^{3})
b. (200pispace cm^{3})
a cylinder has a volume of (500space cm^{3}) and a base area of (25space cm^{2}). what is its height?
a. 25 cm
b. 20 cm
the diameter of a cylinder is 6 cm. what is the radius?
a. 3 cm
b. 12 cm

Explanation:

First Question (Cross - sectional area in Cavalieri's Principle)

Cavalieri's Principle involves comparing the volumes of solids by looking at their cross - sectional areas. A cross - sectional area is the area of a 2 - D slice (like a slice of a loaf of bread) of a 3 - D solid at a particular height. Option a is incorrect as it refers to the length of the base, not an area of a slice.

Step 1: Recall the volume formula for a cylinder

The formula for the volume \(V\) of a cylinder is \(V=A\times h\), where \(A\) is the base area and \(h\) is the height.

Step 2: Substitute the given values

We are given that \(A = 25\pi\space cm^{2}\) and \(h = 10\space cm\). Substituting these into the formula, we get \(V=25\pi\times10\).

Step 3: Calculate the result

\(25\pi\times10 = 250\pi\space cm^{3}\)

Step 1: Recall the volume formula for a cylinder

The formula for the volume \(V\) of a cylinder is \(V = A\times h\), where \(A\) is the base area and \(h\) is the height. We can re - arrange this formula to solve for \(h\): \(h=\frac{V}{A}\).

Step 2: Substitute the given values

We are given that \(V = 500\space cm^{3}\) and \(A = 25\space cm^{2}\). Substituting these into the formula for \(h\), we get \(h=\frac{500}{25}\).

Step 3: Calculate the result

\(\frac{500}{25}=20\space cm\)

Answer:

b. The area of a two - dimensional slice of a solid at a given height

Second Question (Volume of a Cylinder)