Question

use the diagram to help you match each expression in column a to the corresponding term in column b.
column a

1. $2\pi x$
2. $y$
3. $z$
4. $\pi x^2$
5. $\frac{1}{3}\pi x^2 y$

column b
slant height
area of base
volume of cone
circumference of base
height

Explanation:

Step1: Recall formulas

Recall the formula for the circumference of a circle \(C = 2\pi r\) (where \(r=x\) here), the formula for the area of a circle \(A=\pi r^{2}\) (where \(r = x\)), the formula for the volume of a cone \(V=\frac{1}{3}\pi r^{2}h\) (where \(r=x\) and \(h = y\)), and the definitions of height (\(y\)) and slant height (\(z\)) of a cone.

Step2: Match expressions

• For \(2\pi x\): Since the circumference of a circle \(C = 2\pi r\) and \(r=x\), \(2\pi x\) is the circumference of the base.
• For \(y\): By the definition of the height of a cone (perpendicular distance from the base to the vertex), \(y\) is the height.
• For \(z\): By the definition of the slant height of a cone (distance from the vertex to a point on the circular edge of the base), \(z\) is the slant height.
• For \(\pi x^{2}\): Since the area of a circle \(A=\pi r^{2}\) and \(r = x\), \(\pi x^{2}\) is the area of the base.
• For \(\frac{1}{3}\pi x^{2}y\): Since the volume of a cone \(V=\frac{1}{3}\pi r^{2}h\) with \(r=x\) and \(h = y\), \(\frac{1}{3}\pi x^{2}y\) is the volume of the cone.

Answer:

1. \(2\pi x\) - circumference of base; 5. \(y\) - height; 6. \(z\) - slant height; 7. \(\pi x^{2}\) - area of base; 8. \(\frac{1}{3}\pi x^{2}y\) - volume of cone.