Problem 55 - Algebra Form and Function

Question

The volume of a cone of base radius \(r\) and height \(h\) is \((1 / 3) \pi r^{2} h,\) and the volume of a sphere of radius \(r\) is \((4 / 3) \pi r^{3}\). Suppose a particular sphere of radius \(r\) has the same volume as a particular cone of base radius \(r\). (a) Write an equation expressing this situation. (b) What is the height of the cone in terms of \(r\) ?

Step-by-Step Solution

Answer

Answer: The height of the cone is 4 times the radius, or h = 4r.

Step 1: (a) Write an equation expressing this situation

Given that the volume of the cone and sphere are equal, we can write an equation using the volume formulas of a cone and a sphere: \((1 / 3) \pi r^{2} h = (4 / 3) \pi r^{3}.\)

Step 2: (b) Solve for the height of the cone in terms of r

We can now solve for the height h of the cone in terms of the radius r: 1. First, we can divide both sides of the equation by \(\pi r^{2}\) to cancel out the common terms: \(\frac{(1 / 3) \pi r^{2} h}{\pi r^{2}} = \frac{(4 / 3) \pi r^{3}}{\pi r^{2}}\) This simplifies to: \(h = 4r\) 2. The height of the cone `h` can now be easily expressed in terms of the radius `r`: \(h = 4r\) So, the height of the cone is 4 times the radius.

Key Concepts

Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex. In mathematics, when we calculate the volume of a cone, we determine how much space this cone occupies. The key elements of a cone include its base radius (\(r\)) and its height (\(h\)).

• The formula for the volume is: \( \frac{1}{3} \pi r^2 h \).
• The factor \(\frac{1}{3}\) indicates that a cone's volume is one-third of the volume of a cylinder that shares the same base and height.

Understanding this concept becomes crucial when comparing the volumes of cones and other shapes like spheres. In such comparisons, algebraic manipulation helps to derive relationships between the dimensions of the shapes, such as height in terms of radius.

Sphere

A sphere is a perfectly symmetrical three-dimensional shape, where every point on its surface is equidistant from its center. It's a common shape studied in geometry, and it's important to learn how we calculate its volume.

• The volume of a sphere depends only on its radius (\(r\)) and is given by the formula: \( \frac{4}{3} \pi r^3 \).
• The term \(4/3\) adjusts the volume for the spherical shape, considering how all radii extend equally in all directions.

When a sphere and a cone have equal volumes, it sets up a unique situation where we can explore their geometric relationships more deeply. By knowing the algebraic formula for their volumes, we can equate and solve for other variables, like the height of the cone in this scenario.

Algebraic Equations

Algebraic equations are essential tools in mathematics for expressing relationships between variables. In the context of determining the height of a cone when its volume matches that of a sphere, algebraic manipulation is key.

• We set up an equation equating the cone's volume formula (\( \frac{1}{3} \pi r^2 h \)) with that of the sphere (\( \frac{4}{3} \pi r^3 \)).
• The next step involves simplifying the equation by canceling out common terms (like \(\pi\) and \(r^{2}\)).
• Finally, we solve for the desired variable, `h` in this case, showing \(h = 4r\), which means the cone's height is four times the sphere's radius.

This process illustrates how algebraic techniques can be used to derive meaningful conclusions and emphasize the interconnection of geometric and algebraic concepts.