Question

problem 6
this table shows the radiuses of four cones with a height of 18 meters.
radius (m) | 1 | 2 | 3 | 4
volume (cu. m) | 6π | 24π | 54π | 96π
based on your table, is the relationship between the radius of a cone and its volume linear?
yes
no

Explanation:

Step1: Recall Cone Volume Formula

The volume formula for a cone is \( V = \frac{1}{3}\pi r^2 h \). Given \( h = 18 \) m, substitute \( h \) into the formula: \( V=\frac{1}{3}\pi r^2\times18 = 6\pi r^2 \).

Step2: Check Linear Relationship

A linear relationship has the form \( y = mx + b \) (proportional if \( b = 0 \), \( y=mx \)). Here, \( V = 6\pi r^2 \), which is a quadratic (degree 2) relationship, not linear (degree 1). We can also check ratios:

• For \( r = 1 \), \( V = 6\pi \); ratio \( \frac{V}{r}=\frac{6\pi}{1}=6\pi \)
• For \( r = 2 \), \( V = 24\pi \); ratio \( \frac{V}{r}=\frac{24\pi}{2}=12\pi

eq6\pi \)
Since the ratio \( \frac{V}{r} \) is not constant, the relationship is not linear.

Answer:

NO