Question

If the total surface area of a cone is \(384\pi \, cm^2\) and its height is 16 cm, then volume of the cone, in \(cm^3\) is

• A. \(768\pi\)
• B. \(1254\pi\)
• C. \(2304\pi\)
• D. \(3128\pi\)

Hint:

Whenever the height and total surface area of a cone are given, first try identifying a Pythagorean triple for \(r\), \(h\), and \(l\). Here \(12,16,20\) forms a standard triple, making the calculation very easy.

Answer 1

The Correct Option is A

Concept: For a cone, \[ \text{Total Surface Area}=\pi r(r+l) \] where \(r\) is the radius and \(l\) is the slant height. Also, \[ l=\sqrt{r^2+h^2} \] and the volume of a cone is \[ V=\frac{1}{3}\pi r^2h. \]

Step 1: Use the given total surface area.
Given, \[ \pi r(r+l)=384\pi \] Cancelling \(\pi\), \[ r(r+l)=384 \] Also height, \[ h=16 \text{ cm} \] Therefore, \[ l=\sqrt{r^2+16^2} =\sqrt{r^2+256} \] Substituting, \[ r\left(r+\sqrt{r^2+256}\right)=384 \]

Step 2: Determine the radius.
Testing the value \(r=12\), \[ l=\sqrt{12^2+16^2} =\sqrt{144+256} =\sqrt{400} =20 \] Now, \[ r(r+l) = 12(12+20) = 12\times 32 = 384 \] Hence, \[ r=12 \text{ cm} \] and \[ l=20 \text{ cm}. \]

Step 3: Calculate the volume of the cone.
Using \[ V=\frac13\pi r^2h \] \[ V = \frac13\pi(12)^2(16) \] \[ = \frac13\pi(144)(16) \] \[ = \frac{2304}{3}\pi \] \[ = 768\pi \] Therefore, \[ \boxed{768\pi \; cm^3} \]