Problem 80 - Algebra A Combined Function

Question

The formula for the radius \(r\) of a cone with height 7 centimeters and volume \(V\) is $$ r=\sqrt{\frac{3 V}{7 \pi}} $$ Rationalize the numerator of the radical expression in this formula.

Step-by-Step Solution

Answer

The rationalized form is \( \frac{\sqrt{3V\pi}}{\pi\sqrt{7}} \).

Step 1: Understanding the Formula

The given formula is \( r = \sqrt{\frac{3V}{7\pi}} \), where \( r \) is the radius of the cone, \( V \) is the volume, and \( \pi \) is a constant (approx. 3.14159). We need to rationalize the numerator of the expression under the square root.

Step 2: Identifying the Radical Expression

Currently, the expression under the square root is \( \frac{3V}{7\pi} \). To rationalize, we need to modify this to remove any irrational numbers (specifically \( \pi \) here) from the denominator of the expression under the radical.

Step 3: Multiplying to Rationalize

Multiply both the numerator and the denominator of the expression \( \frac{3V}{7\pi} \) by \( \pi \). This will give \( \frac{3V \cdot \pi}{7\pi \cdot \pi} = \frac{3V\pi}{7\pi^2} \). Now, we have a rationalized expression under the radical.

Step 4: Simplifying the Expression

The expression is now \( \sqrt{\frac{3V\pi}{7\pi^2}} \). To simplify, factor out \( \frac{1}{\pi} \) from the square root as \( \frac{1}{\sqrt{\pi}} \), which leads to: \( \frac{\sqrt{3V\pi}}{\pi\sqrt{7}} \). Since additional steps to simplify further are not provided, this is our rationalized form.

Key Concepts

Rationalizing Denominators

Rationalizing the denominator is a technique in algebra to eliminate irrational numbers from the denominator of a fraction. When a fraction has an irrational number such as a square root, it can make calculations cumbersome. To make things more straightforward, mathematicians prefer to "rationalize" these forms.

• To rationalize a denominator containing a square root, multiply both the numerator and the denominator by a form of 1 that will remove the square root from the denominator.
• For instance, if the denominator is \( \sqrt{a} \), you can multiply the fraction by \( \frac{\sqrt{a}}{\sqrt{a}} \) to rationalize it.
• In our original problem, we have \( \frac{3V}{7\pi} \). By multiplying numerator and denominator by \( \pi \), we aim to get rid of \( \pi \) in the denominator of the resulting expression under the square root.

This adjustment results in an expression that's often easier to use in further computations, both in physical contexts and mathematical derivations.

Volume of a Cone

The volume of a cone is a three-dimensional space measurement that can be calculated using the formula:\[V = \frac{1}{3} \pi r^2 h\]where \( V \) is the volume, \( \pi \) is a constant (approximately 3.14159), \( r \) is the radius of the base, and \( h \) is the height of the cone. This formula comes from the fact that a cone is essentially a pyramid with a circular base.

• The cone volume formula is derived from that of a cylinder \( \pi r^2 h \), which is the full base area times height, divided by three to account for the tapering shape of cones.
• Understanding this concept is crucial when analyzing the relationship between the radius, height, and volume of a cone.
• So, if you have two of these three measurements, you can rearrange the formula to solve for the missing dimension, as seen in our problem: solving for \( r \) given height and volume.

Having a strong grasp of this formula is useful for various real-world applications such as architecture and engineering where conical shapes are prevalent.

Radical Expressions

Radical expressions involve roots, like square roots or cube roots, and handle values both within the root sign and outside it. They are found in various mathematical contexts and seem a bit tricky at first.

• Radical expressions often need simplification, particularly when they contain fractions, to make them more understandable and usable.
• For example, \( \sqrt{ \frac{a}{b} } \) can be separated as \( \frac{\sqrt{a}}{\sqrt{b}} \), which sometimes assists in simplifying the expression further, just as in rationalizing denominators.
• In our exercise, the radical expression \( \sqrt{\frac{3V}{7\pi}} \) becomes \( \frac{\sqrt{3V\pi}}{\pi\sqrt{7}} \) after rationalization. Here, you can tackle the radical separately by multiplying by \( \pi \), which results in the further simplified and more manageable situation.

Handling radical expressions is all about making calculations manageable and understandable. Mastering these manipulations is crucial for success in algebra and beyond.