Question

A solid sphere is melted and recast into a right circular cone with a base radius equal to the radius of the sphere. What is the ratio of the height and radius of the cone so formed? (a) \(4: 3\) (b) \(2: 3\) (c) \(3: 4\) (d) none of these

Short answer

a) 2:1 b) 3:1 c) 1:1 d) None of these Answer: d) None of these. Explanation: By equating the volumes of both the sphere and the cone and solving for the ratio of height to radius, we find that the ratio is 4:1, which is not among the given options.

Step-by-step solution

Volume formula for a sphere and a cone

First, let's start with the volume formulas for a sphere and a cone. - The volume of a sphere with radius r is given by the formula: \(V_{sphere} = \frac{4}{3}\pi r^3\). - The volume of a cone with radius r and height h is given by the formula: \(V_{cone} = \frac{1}{3}\pi r^2h\).

Set up an equation relating the volumes

Since the sphere is melted and recast into the cone, the volumes of both shapes should be equal. We can set up the equation as follows: $$V_{sphere} = V_{cone}.$$ So, $$\frac{4}{3}\pi r^3 = \frac{1}{3}\pi r^2h.$$

Solve for the ratio of height to radius

Now, we need to find the ratio \(\frac{h}{r}\). We can simplify our equation by multiplying both sides by 3 and canceling the common terms: $$4\pi r^3 = \pi r^2h.$$ Next, divide both sides of the equation by \(\pi r^2\): $$4r = h.$$ Now, we can find the ratio \(\frac{h}{r}\). $$\frac{h}{r} = \frac{4r}{r} = 4.$$ Therefore, the ratio of the height to the radius of the cone formed is 4:1, which is not one of the given options. So, the answer is (d) none of these.

Key concepts

Volume of Sphere

Understanding the volume of a sphere is essential in geometry, as spheres are a fundamental geometric shape. A sphere is perfectly round with every point on its surface equidistant from its center, much like a ball.

The formula to calculate the volume of a sphere is given by:

• \[ V_{sphere} = \frac{4}{3}\pi r^3 \]

Here, \( r \) represents the radius of the sphere, which is the distance from the center of the sphere to any point on its surface.

This formula, \( \frac{4}{3}\pi r^3 \), involves π (Pi), a natural mathematical constant approximately equal to 3.14159. The volume of the sphere is directly proportional to the cube of its radius, making the radius a critical factor in determining the space occupied by the sphere. By understanding this, you can solve various problems involving spheres, including those that require comparing the volume of a sphere with other shapes, such as a cone.

Volume of Cone

The volume of a cone is equally fascinating and understanding its calculation can help in a multitude of mathematical challenges. A cone is basically a pyramid with a circular base, tapering smoothly to a point, known as its apex. This shape can be visualized like an ice cream cone or a funnel.

The volume of a cone is calculated using the formula:

• \[ V_{cone} = \frac{1}{3}\pi r^2h \]

In this formula, \( r \) is the radius of the circular base, and \( h \) is the height of the cone, measured from the base to the apex. Notice the appearance of π (Pi) and the fraction \( \frac{1}{3} \), which signifies that the volume of a cone is one-third the product of the area of the base and the height. This fraction arises from the integration process in calculus that derives the formula.

This formula highlights how both the radius and height of the cone significantly impact its total volume. It also sets up interesting scenarios like the one in our exercise, where the volume from a different shape like a sphere is recast into a cone, requiring an understanding of these fundamental formulas to solve.

Ratio in Geometry

Ratios are pivotal in geometry, providing a powerful way to compare two quantities of the same kind. In geometry exercises, understanding and applying ratios helps bridge different dimensions, such as comparing aspects like height to radius in different geometrical shapes.

For example, in our exercise, we deal with a solid sphere melted and recast into a cone with the same base radius. We then determine the ratio of the height to the radius of this newly formed cone. This is done through a series of mathematical manipulations which simplify the equality of volumes. Understanding these relationships in terms of ratios can simplify problem-solving, help identify similarities, and draw comparisons between different geometrical entities more efficiently. When solving ratio-related problems, it's important to keep consistent units and accurate measurement to ensure the ratio reflects the correct relationship. Often, these ratios can be reduced to simple forms which are easier to interpret and apply in further geometry problems.

Mathematical Problem Solving

Mathematical problem solving is a critical skill that involves identifying the problem, strategizing, and applying mathematical principles to come up with a solution. It's not just about reaching an answer but doing so in a structured and logical manner.

In our exercise, we begin by utilizing known volume formulas of a sphere and a cone to establish a connection between the two shapes. Through deduction, we set the volumes equal to each other, then manipulate the equation to identify the height-to-radius ratio of the resulting cone. This process relies on several problem-solving skills including:

• Recognizing the type of mathematical problem
• Applying the appropriate formulas and mathematical concepts
• Simplifying complex equations to solve for unknowns

By following such systematic approaches, you can more easily handle a wide range of mathematical problems. Effective problem solving also involves verifying your results to ensure correctness and making sure that all possibilities are covered, especially when the options do not match the expected solution, as in the provided exercise.