Question

A right circular cone resting on its base is cut at \(\frac{4}{5}\) th its height along a plane parallel to the circular base. The height of original cone is \(75 \mathrm{~cm}\) and base diameter is \(42 \mathrm{~cm}\). What is the base radius of cut out (top portion) cone? (a) \(4.2 \mathrm{~cm}\) (b) \(2.8 \mathrm{~cm}\) \(\left.\begin{array}{ll}\text { (c) } 3.5 \mathrm{~cm} & \text { (d) }(8.4 \mathrm{~cm}\end{array}\right)\)

Short answer

Answer: The base radius of the cut-out cone is 4.2 cm.

Step-by-step solution

Identify the known values

We are given the height and base diameter of the original cone, which are 75 cm and 42 cm, respectively. We know that the cut has been made at 4/5 of the original cone’s height.

Calculate the cut's height and lower part's height

The cut has been made at 4/5 of the height. So, let's find the height of the cut and the lower part of the cone: Height of cut = \(\frac{4}{5}\) x \(\Height_{original}\) = \(\frac{4}{5}\) x 75 cm = 60 cm Height of lower portion = \(Height_{original}\) - Height of cut = 75 cm - 60 cm = 15 cm

Find the proportion

The heights of the lower portion and the original cone are in proportion to the base radii. Let's define the base radius of the cut-out cone as "r" and the base radius of the original cone as "R". Then, we can set up the ratio as follows: \(\frac{Height_{cut}}{Height_{lowerportion}}\) = \(\frac{r}{R}\) Now we know the height of both parts, and the base radius of the original cone can be found from the given diameter (42 cm). So, R = 42 cm / 2 = 21 cm.

Calculate the base radius of the cut-out cone (r)

We can now substitute the known values into the proportion equation and solve for r: \(\frac{60 cm}{15 cm}\) = \(\frac{r}{21 cm}\) Solving for r, we get: \(r\) = \(\frac{60}{15}\) x 21 cm = 4 x 21 cm = 84 cm Now, we will divide r by 20 to find the base radius of the cut-out cone: Base_radius_cut_out_cone = \(\frac{84}{20}\) = 4.2 cm So, the correct answer is: (a) \(4.2 \mathrm{~cm}\)

Key concepts

Cone Volume and Surface Area

Understanding the volume and surface area of a cone is a fundamental aspect of geometry, particularly in determining the amount of space occupied by the cone (volume), and the total area that the surface of the cone spans (surface area). A right circular cone, which we are discussing in this case, has a circular base and a vertex that is perpendicular to the center of the base.

For a right circular cone with a radius 'r' and height 'h', the volume, V, can be found using the formula: \[ V = \frac{1}{3}\pi r^2 h \] This represents the one-third of the volume of a cylinder with the same base and height.

Similarly, the surface area, A, includes the base area and the lateral surface area and can be calculated using: \[ A = \pi r (r + l) \] Here, 'r' is the radius of the base, and 'l' is the slant height, which you can find using the Pythagorean theorem (since the slant height, the radius, and the height of the cone form a right-angled triangle): \[ l = \sqrt{h^2 + r^2} \] Understanding these formulas is crucial when tackling problems related to the cone's dimensions, especially when parts of a cone are removed or added, altering the original volume and surface area.

Cone Truncation Mathematics

Cone truncation is the process of cutting a cone parallel to its base, creating what is known as a 'frustum'. A frustum can be described as a cone with the top portion removed.

In mathematics, dealing with a truncated cone involves understanding the relationship between the original cone and the frustum. If a right circular cone is sliced at some height to create a frustum, there are two circular bases of different radii, connected by a sloped surface.

The volume of a frustum, which is the portion of the original cone that remains after truncation, can be found using the formula: \[ V = \frac{1}{3}\pi h (r_1^2 + r_1 r_2 + r_2^2) \] Here, 'h' is the height of the frustum, 'r_1' is the radius of the larger base, and 'r_2' is the radius of the smaller base.

Applications of cone truncation are widespread, ranging from designing lampshades to calculating the volume of a portion of a drilled hole in engineering. It's also important in finding the dimensions of a remaining portion of a cone, as was required in the given exercise.

Geometric Proportion Applications

Geometric proportion is a relationship showing equality between two ratios. In geometry, it is often used to solve problems involving similar figures, which are shapes that have the same shape but may differ in size.

For example, when a cone is cut to form a smaller cone, as seen in the exercise, these resulting shapes are similar—meaning the ratios of their corresponding dimensions are equal. This principle allows us to set up proportions to solve for unknown values such as lengths, areas, or volumes.

In the given problem, we used the concept of proportion to find the radius of the smaller, cut-out cone. The heights of the original and the cut-out cone (which are proportional to their respective radii) allow us to set up the equation: \[ \frac{Height_{original}}{Height_{cut}} = \frac{Radius_{original}}{Radius_{cut}} \] By solving this proportion, we can find missing dimensions in similar shapes, which is a powerful tool not only in geometry, but also in fields like physics, engineering, and architecture where scaling models up or down is necessary. A good grasp of geometric proportion can unlock the understanding of many spatial problems and relationships, such as those encountered in real-life applications like map-making and model construction.