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# A right circular cone has a base radius r cm and a vertical angle 2yo. The heigh...

### Question

A right circular cone has a base radius r cm and a vertical angle 2yo. The height of the cone is

A)
r tan yo cm
B)
r sin yo cm
C)
r cot yo cm
D)
r cos yo cm
E)
r cosec yo cm

### Explanation:

A right circular cone has a base radius $$r$$ cm and a vertical angle $$2y°$$. We are asked to find the height of the cone. First, let's visualize the cone and its properties.

Imagine the cone with its base on the ground and vertex pointing up. The vertical angle ($$2y°$$) is the angle formed between a line from the center of the base to a point on the circumference and another line from the vertex to the same point on the circumference. This is an isosceles triangle, with two equal sides being the slant height of the cone. We can find the height by bisecting the vertical angle ($$2y°$$) at the vertex, which divides the isosceles triangle into two right-angled triangles.

In each of these right-angled triangles, the base is the radius ($$r$$) of the cone, the height ($$h$$) is what we need to find, and the angle between the base and height is $$y°$$. We can use trigonometry to find the height. Since we have the angle ($$y°$$) and the adjacent side ($$r$$), we can use the tangent function.

$\tan{y°} = \frac{h}{r}$

To find the height ($$h$$), we can rearrange the equation:

$h = r \tan{y°}$

However, we made an error in the question. The correct option should be $$r \tan{y°}$$ cm, which is not present in the given options. The correct answer should be Option A, but it should read: Option A: $$r \tan{y°}$$ cm.

Therefore, the height of the cone is $$r \tan{y°}$$ cm.

## Discussion (2)

• In a right circular cone, the height can be found using the radius and the half of the vertical angle (which is the angle between the slant height and the base radius) using the tangent trigonometric function.

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this case, the height is the opposite side and the radius is the adjacent side to the angle $$yo$$.

So, the height $$h$$ of the cone can be expressed as:

$h = r \tan y^o$

• $$\frac{r}{h}$$ = tan yo
h = $$\frac{r}{tan y^o}$$