Question

which factor directly explains why the volume of a cone is smaller than that of a cylinder with the same base and height?
a. a cone has a smaller base.
b. a cone tapers to a point, taking up less space.
c. a cone has a smaller height.
d. all cones are smaller than all cylinders.

how many cones would it take to fill a cylinder with the same base and height?
a. three cones
b. two cones
c. one cone
d. four cones

a cone has a radius of 7 cm and a height of 10 cm. what is its volume?
a. 490π cm³
b. 154π cm³
c. 163.33π cm³
d. 245π cm³

what is the formula for the volume of a cylinder?
a. πrl²
b. 1/3 πr²h
c. 2πrh
d. πr²h

Explanation:

First Question

When a cone and a cylinder have the same base and height, the cone's shape (tapering to a point) means it occupies less space. The volume formula of a cone \(V_{cone}=\frac{1}{3}\pi r^{2}h\) and for a cylinder \(V_{cylinder}=\pi r^{2}h\) (where \(r\) is the base radius and \(h\) is the height). Since the base (\(r\)) and height (\(h\)) are the same, the factor is the shape (tapering).

Using the volume formulas \(V_{cone}=\frac{1}{3}\pi r^{2}h\) and \(V_{cylinder}=\pi r^{2}h\) (same \(r\) and \(h\)). If we let \(n\) be the number of cones to fill the cylinder, then \(n\times V_{cone}=V_{cylinder}\). Substituting the formulas: \(n\times\frac{1}{3}\pi r^{2}h=\pi r^{2}h\). Solving for \(n\) (divide both sides by \(\frac{1}{3}\pi r^{2}h\)), we get \(n = 3\).

Step1: Recall the volume formula for a cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\), where \(r = 7\space cm\) and \(h=10\space cm\).

Step2: Substitute the values into the formula

Substitute \(r = 7\) and \(h = 10\) into \(V=\frac{1}{3}\pi r^{2}h\). We get \(V=\frac{1}{3}\pi\times(7)^{2}\times10=\frac{1}{3}\pi\times49\times 10=\frac{490}{3}\pi\approx163.33\pi\space cm^{3}\)

Answer:

b. A cone tapers to a point, taking up less space.

Second Question