A lamp shade is in the shape of a part of a cone and its top and bottom ends are circles whose circumferences are respectively 30 cm and 40 cm. The perpendicular distance between the ends is 6 cm. If the cone were to be completed, then how far would its vertex be from the top end?
18 cm
Understanding the Lamp Shade as a Cone Frustum
The lamp shade described is in the shape of a frustum of a cone. A frustum is what remains of a cone after a smaller cone is removed from the top by cutting parallel to the base. We are given the circumferences of the top and bottom circular ends and the perpendicular distance (height) between these ends. The question asks for the distance from the vertex of the completed cone to the top end of the lamp shade frustum.
Calculating Radii from Circumferences
The circumference of a circle is given by the formula \(C = 2\pi r\), where \(C\) is the circumference and \(r\) is the radius.
- Bottom end: Circumference \(C_b = 40\) cm. Let the radius of the bottom end be \(R\). \(40 = 2\pi R\) \(R = \frac{40}{2\pi} = \frac{20}{\pi}\) cm
- Top end: Circumference \(C_t = 30\) cm. Let the radius of the top end be \(r\). \(30 = 2\pi r\) \(r = \frac{30}{2\pi} = \frac{15}{\pi}\) cm
Using Similar Triangles to Find the Vertex Distance
When the cone is completed, we can imagine a vertical cross-section through the axis of the cone. This cross-section will show two similar triangles:
- A larger triangle representing the complete cone, with base radius \(R\) and total height \(H\).
- A smaller triangle representing the cone that was removed from the top, with base radius \(r\) and height \(h\) (which is the distance from the vertex to the top end of the frustum).
The height of the frustum is the difference between the total height of the large cone (\(H\)) and the height of the smaller cone (\(h\)). We are given the perpendicular distance between the ends (frustum height) as 6 cm.
So, \(H - h = 6\) cm.
Because the two triangles are similar, the ratio of their corresponding sides is equal. Specifically, the ratio of their radii is equal to the ratio of their heights:
\( \frac{R}{r} = \frac{H}{h} \)Now, substitute the values of \(R\) and \(r\) we calculated:
\( \frac{\frac{20}{\pi}}{\frac{15}{\pi}} = \frac{H}{h} \) \( \frac{20}{15} = \frac{H}{h} \)Simplify the fraction:
\( \frac{4}{3} = \frac{H}{h} \)This gives us a relationship between \(H\) and \(h\):
\( H = \frac{4}{3}h \)Solving for the Distance from Vertex to Top End
We have two equations involving \(H\) and \(h\):
- \(H - h = 6\)
- \(H = \frac{4}{3}h\)
Substitute the expression for \(H\) from equation (2) into equation (1):
\( \left(\frac{4}{3}h\right) - h = 6 \) \( \frac{4}{3}h - \frac{3}{3}h = 6 \) \( \left(\frac{4}{3} - 1\right)h = 6 \) \( \frac{1}{3}h = 6 \)Now, solve for \(h\):
\( h = 6 \times 3 \) \( h = 18 \)The value of \(h\) represents the height of the smaller cone that was removed, which is exactly the distance from the vertex of the completed cone to the top end of the lamp shade frustum.
Therefore, the vertex would be 18 cm away from the top end.
| Given Information | Calculated Values | Formula Used |
|---|---|---|
| Bottom Circumference = 40 cm | Bottom Radius (R) = \(20/\pi\) cm | \(C = 2\pi r\) |
| Top Circumference = 30 cm | Top Radius (r) = \(15/\pi\) cm | \(C = 2\pi r\) |
| Frustum Height = 6 cm | Vertex to Top Height (h) = 18 cm | Similar Triangles (\(R/r = H/h\)) and \(H-h=\) Frustum Height |
Revision Table: Key Concepts for Frustum Problems
| Concept | Description | Relevance to Problem |
|---|---|---|
| Cone Frustum | Part of a cone between two parallel planes. | Describes the shape of the lamp shade. |
| Circumference | Distance around a circle (\(2\pi r\)). | Used to find the radii of the ends. |
| Similar Triangles | Triangles with the same shape but different sizes. Corresponding angles are equal, and corresponding sides are proportional. | Used to relate the radii and heights of the large and small cones. |
| Height of Frustum | Perpendicular distance between the parallel ends. | Provides one equation for solving for heights (\(H-h\)). |
Additional Information: Cone Geometry and Similar Solids
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. When a cone is cut by a plane parallel to its base, the resulting two parts are a smaller cone and a frustum.
The principle of similar triangles is fundamental in solving problems involving cones and frustums when dealing with heights and radii. If you imagine cutting a cone vertically through its vertex, the resulting shape is an isosceles triangle. If you make a parallel cut (which forms the frustum), this cut corresponds to a line segment parallel to the base of the triangle. This parallel line segment cuts off a smaller triangle at the top that is similar to the original large triangle.
The ratio of corresponding linear dimensions (like radii, heights, or slant heights) in similar cones (or the cone and the smaller cone removed from the top) is constant. The ratio of corresponding areas (like base areas or lateral surface areas) is the square of the ratio of linear dimensions. The ratio of corresponding volumes is the cube of the ratio of linear dimensions. This problem primarily uses the ratio of linear dimensions (radii and heights).
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