Write each formula as an English phrase using the word varies or proportional. V = 1/3 πr2h, where V is the volume of a cone of radius r and height h

Ch. 3 - Polynomial and Rational Functions

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Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 25

Chapter 4, Problem 25

Write each formula as an English phrase using the word varies or proportional. V = 1/3 πr²h, where V is the volume of a cone of radius r and height h

Verified step by step guidance

Identify the variables in the formula: \(V\) represents the volume of the cone, \(r\) is the radius, and \(h\) is the height.

Recognize the constants and coefficients in the formula: \(\frac{1}{3}\) and \(\pi\) are constants that do not change with \(r\) or \(h\).

Observe how \(V\) depends on \(r\) and \(h\): \(V\) is multiplied by \(r^2\) and \(h\), indicating that \(V\) varies with both \(r^2\) and \(h\).

Express the relationship using the word 'varies' or 'proportional': Since \(V\) is equal to a constant times \(r^2\) times \(h\), we say that \(V\) varies jointly as the square of the radius and the height.

Write the English phrase: 'The volume \(V\) of a cone varies jointly as the square of its radius \(r\) and its height \(h\).'

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Direct Variation

Direct variation describes a relationship where one quantity changes proportionally with another. If y varies directly as x, then y = kx for some constant k. In the formula for volume, understanding which variables increase or decrease together helps express the relationship using 'varies' or 'proportional'.

Volume of a Cone Formula

The volume of a cone is given by V = (1/3)πr²h, where r is the radius and h is the height. This formula shows how volume depends on both the square of the radius and the height, indicating that volume changes with these dimensions. Recognizing this helps translate the formula into a verbal proportional relationship.

Translating Mathematical Expressions into Words

Translating formulas into English involves expressing mathematical relationships clearly using terms like 'varies directly' or 'is proportional to'. This skill requires identifying constants and variables and describing how one quantity depends on others, which aids in understanding and communicating algebraic relationships.

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