Question

Charles has a right circular cylinder which he inserted completely into a right circular cone of height \(30 \mathrm{~cm}\). The vertical angle of the cone is \(60^{\circ}\) and the diameter of the cylinder is \(8 \sqrt{3} \mathrm{~cm}\). What is the volume of the cone? (a) \(\frac{3000}{7} \pi \mathrm{cm}^{3}\) (b) \(3000 \pi \mathrm{cm}^{3}\) (c) \(4860 \pi \mathrm{cm}^{3}\) (d) can't be determind

Short answer

(We can assume that the angles are given with respect to the center of the cone's base.) a. \(\frac{960\pi}{3} \mathrm{~cm}^3\) b. \(288\pi \mathrm{~cm}^3\) c. \(\frac{216\pi}{3} \mathrm{~cm}^3\) d. It can't be determined with the given options. Answer: d. It can't be determined with the given options.

Step-by-step solution

Find the radius of the cone

To find the radius of the cone, we will use the information about the diameter of the cylinder and the vertical angle of the cone. Since the diameter of the cylinder is \(8\sqrt{3} \mathrm{~cm}\), the radius of the cylinder is \(\frac{8\sqrt{3}}{2}=4\sqrt{3} \mathrm{~cm}\). Since the vertical angle of the cone is \(60^{\circ}\), it means that the lateral sides of the cone make an angle of \(30^{\circ}\) with the vertical axis. By trigonometry, we get that one-half of the diameter has a length of \(\frac{30 \mathrm{~cm}}{\tan(30^{\circ})}=4\sqrt{3} \mathrm{~cm}\), which is equal to the radius of the cylinder. So, the radius of the cone is: \(r=4\sqrt{3}~\mathrm{cm}\)

Find the volume of the cone using its radius and height

Now that we have the radius and the height of the cone, we can use the formula for the volume of a cone: \(V_\text{cone}=\frac{1}{3}\pi r^2h\) Substituting the values we've found for the radius and the height: \(V_\text{cone}=\frac{1}{3}\pi (4\sqrt{3}~\mathrm{~cm})^2(30~\mathrm{~cm})\)

Calculate the volume of the cone

Simplify the expression for the cone's volume: \(V_\text{cone}=\frac{1}{3}\pi (48~\mathrm{~cm}^2)(30~\mathrm{~cm}) = \frac{1}{3}\pi (1440~\mathrm{~cm}^3)= 480\pi \mathrm{~cm}^3\) The volume of the cone is: \(V_\text{cone}=480\pi \mathrm{~cm}^3\) Unfortunately, none of the given answers (a, b, c, or d) matches the found value for the volume of the cone. So the correct answer must be (d) it can't be determined with the given options.

Key concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It’s particularly useful when dealing with right-angled triangles, where one angle is exactly 90 degrees. In our exercise, we applied trigonometry to find the radius of the cone using the vertical angle and the height of the cone.

The process involves understanding trigonometric ratios such as sine, cosine, and tangent. The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side. Here, we used the tangent function because we have an angle (\(30^{\text{o}}\)) and we want to find the opposite side (radius of the cone), given the adjacent side (the height of the cone).

The formula \(\tan(30^{\text{o}}) = \frac{\text{opposite side}}{\text{adjacent side}}\) helped us conclude that the radius of the cone, which is also the radius of the cylinder since it fits perfectly inside the cone, is \(4\sqrt{3} \mathrm{~cm}\). By mastering such trigonometric concepts, students can solve a variety of problems in geometry, physics, and engineering.

Geometry

Geometry is the part of mathematics that deals with shapes, sizes, and the properties of space. It enables us to comprehend the physical world through shapes and figures. One fundamental concept in geometry is the ability to calculate the volume of various solid figures like cones, cylinders, spheres, etc.

In the problem at hand, we deal with two geometrical shapes: a right circular cylinder and a right circular cone. Their relationship is vital to solving the exercise. Understanding the properties of these shapes, such as how the dimensions of one can determine aspects of the other, is essential to geometry. When teaching or presenting geometric solutions, using clear visual aids and step-by-step explanations can greatly assist students in grasping these sometimes abstract concepts.

Circular Cylinder

A circular cylinder is a three-dimensional shape with two parallel circular bases and a curved surface connecting them. The volume of a cylinder can be found using the formula \(V = \(pi\)r^2h\), where \(r\) is the radius of the bases and \(h\) is the height of the cylinder.

In the problem we are discussing, we used the diameter of the cylinder to find the radius. It's important for students to remember that the diameter is twice the radius. By understanding this, they can solve for the radius directly and use it in further calculations. Another key point is that the cylinder fits entirely inside the cone, indicating that both the base of the cylinder and the base of the cone have equal radii. This concept is crucial for determining the cone's dimensions and its volume.

Right Circular Cone

A right circular cone is a solid that has a circular base and a pointed ‘apex’ or vertex that is perpendicular to the base. The volume of a right circular cone is one third the volume of a cylinder with the same base and height, represented by the formula \(V = \frac{1}{3}(pi)r^2h\).

In our particular problem, once we determined the radius of the cone by using trigonometry and the height as given, we could easily compute the cone’s volume. Remember, to calculate the volume of a cone correctly, one must square the radius, multiply by \(pi\), and then by the height, before finally dividing by three. When the formula is applied with precision and the necessary geometrical relations are identified, the volume can be found successfully. However, students must pay close attention to their computation, as a simple mistake in the process could lead to an incorrect result - one that might not even be listed in the answer choices.