Question

Given a conical pile with a semi-vertical angle of 30°, and at time t the radius of the base is r, find the height of the cone in terms of r. Hence, show that the volume V of the cone is:

V=πr³√ 3 /3
A) h=2r√ 3, V=πr³√ 3
B) h=r√ 3, V=πr³√ 3/3
C) h=r√ 3, V=πr³√ 3
D) h=2r√ 3, V=πr³√ 3/3

Answer 1

Let \( h \) be the height of the cone, and \( r \) be the radius of the base. The semi-vertical angle \( \theta \) is given as 30°.

Using trigonometry, we can relate the height, radius, and the semi-vertical angle in the cone:

\[ \tan(\theta) = \frac{r}{h} \]

Given \( \theta = 30° \), we have:

\[ \tan(30°) = \frac{r}{h} \]

\[ \frac{1}{\sqrt{3}} = \frac{r}{h} \]

\[ h = r\sqrt{3} \]

Now, we'll show that the volume \( V \) of the cone is \( \frac{\pi r^3 \sqrt{3}}{3} \):

The volume of a cone is given by the formula:

\[ V = \frac{1}{3} \pi r^2 h \]

Substitute \( h = r\sqrt{3} \) into the formula:

\[ V = \frac{1}{3} \pi r^2 (r\sqrt{3}) \]

\[ V = \frac{\pi r^3 \sqrt{3}}{3} \]

So, the correct option is:

D) \( h = 2r\sqrt{3} \) , \( V = \frac{\pi r^3 \sqrt{3}}{3} \)

Answer 2

The height of the cone can be found using trigonometry by relating the semi-vertical angle to the slant height. The volume of the cone can then be calculated using the formula V = (1/3)πr²h. Option D, h = 2r√3, V = πr³√3/3, is the correct representation of the height and volume of the cone.

Step-by-step explanation:

The height of the cone can be found using trigonometry. In a cone, the semi-vertical angle is half of the angle formed by the cone's axis and one of the slant heights.

Therefore, in this case, the angle formed by the cone's axis and a slant height is 60°. Using trigonometry, we can express the height (h) in terms of the radius (r) as h = 2r√3.

To calculate the volume of the cone, we use the formula V = (1/3)πr²h. Substituting the value of h we derived earlier, the volume simplifies to V = (1/3)πr³√3.

Therefore, the correct option that represents the height and volume of the cone is h = 2r√3, V = πr³√3/3 (Option D).