Mathematics, 30.07.2021 22:40 Liantic34

Suppose the volume of the cone is 324pi Find dy/dx when x=6 and y=27

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Answer from: michaellangley

$\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Equality Properties

Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of Equality

Calculus

Differentiation

DerivativesDerivative Notation

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Derivative Rule [Quotient Rule]: $\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle V = \frac{1}{3} \pi x^2y$

$\displaystyle V = 324 \pi$

$\displaystyle x = 6$

$\displaystyle y = 27$

Step 2: Differentiate

Substitute in volume [Volume Formula]: $\displaystyle 324 \pi = \frac{1}{3} \pi x^2y$ [Equality Properties] Rewrite: $\displaystyle y = \frac{972}{x^2}$ Quotient Rule: $\displaystyle \frac{dy}{dx} = \frac{(972)'x^2 - (x^2)'972}{(x^2)^2}$ Basic Power Rule: $\displaystyle \frac{dy}{dx} = \frac{0x^2 - (2x)972}{(x^2)^2}$ Simplify: $\displaystyle \frac{dy}{dx} = \frac{-1944x}{x^4}$ Simplify: $\displaystyle \frac{dy}{dx} = \frac{-1944}{x^3}$

Step 3: Evaluate

Substitute in variables [Derivative]: $\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = \frac{-1944}{6^3}$ Simplify: $\displaystyle \frac{dy}{dx} \bigg| \limits_{x = 6} = -9$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Differentiation

Answer from: Quest

the answer for this question is 38

Answer from: Quest

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Suppose the volume of the cone is 324pi Find dy/dx when x=6 and y=27
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