Question

the slant height of a cone is given. can you calculate its volume using this information alone?
a. yes
b. none of the answers
c. sometimes
d. no

Explanation:

Step1: Recall the volume formula of a cone

The volume formula of a cone is \(V=\frac{1}{3}\pi r^{2}h\), where \(r\) is the radius of the base and \(h\) is the height of the cone.

Step2: Recall the relationship between slant height \(l\), radius \(r\) and height \(h\)

We know that \(l = \sqrt{r^{2}+h^{2}}\) (by the Pythagorean theorem for the right - triangle formed by the radius, height and slant height of the cone). If only the slant height \(l\) is given, we have two unknowns \(r\) and \(h\) in the equations \(V=\frac{1}{3}\pi r^{2}h\) and \(l = \sqrt{r^{2}+h^{2}}\). We cannot uniquely determine the values of \(r\) and \(h\) (for example, if \(l = 5\), we could have \(r = 3,h = 4\) or \(r=\sqrt{10},h=\sqrt{15}\) etc.). So we cannot calculate the volume of the cone using the slant height alone.

Answer:

D. No