Explanation Given information: It is provided the can has height of 4 centimeters and a radius 2 centimeters where the volume of the paint in a right cylindrical can is provided by V = 4 t 2 − t , where t represents the time in seconds and V represents the volume in the cubic centimeters. Formula used: The steps for solving the related rate problem are: Step1: List the changing and related quantities followed by restating the problem in rate of change of terms. Step 2: Rewrite the problem using for the changing quantities for their derivatives. Step 3: If possible draw the diagram of the changing quantities and find an equation relating the changing quantities. Step 4: Differentiate with respect to the equation (s) in order to get rate of change in relating quantity. Step 5: Now substitute the values of the provided quantity and their derivative into the derived equation and solve them to get answer. Volume of cylinder: The volume of the cylinder with height h and radius r is given by: V = π r 2 h Quadratic formula: The solution of the quadratic equation a x 2 + b x + c = 0 is: x = − b ± b 2 − 4 a c 2 a Calculation: Consider the steps of the formula: The changing quantities are the volume V and height h . Since, the cylindrical can has a radius 2 centimeters and the volume of the paint in a right cylindrical can is given by V = 4 t 2 − t . Thus, the volume V is 4 t 2 − t , the radius is constant at 2 . Thus, d V d t = 8 t − 1 and r = 2 Since, the rate at which the level rising when the height is 2 . Thus, the above sentence can be rephrased as: What is the rate of change of h Thus, d h d t = ? when h = 2 Consider the relationship equation: Consider the formula as: V = π r 2 h Differentiate above equation with respect to t as: d V d t = 2 π r h Substitute 2 for r in above equation to get: d V d t = 2 ( 2 ) π d h d t = 4 π d h d t Thus, d h d t = 1 4 π ⋅ d V d t Substitute 8 t − 1 for d V d t in above equation to get: d h d t = 8 t − 1 4 π Since, the value of t is to be calculated at h = 2

Finite Mathematics and Applied Calculus (MindTap Course List)

7th Edition

ISBN: 9781337274203

Author: Stefan Waner, Steven Costenoble

Publisher: Cengage Learning

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Question

Chapter 12.5, Problem 45E

To determine

To calculate: The rate at which the level rising when the height is 2 centimeters and the can has height of 4 centimeters and a radius 2 centimeters where the volume of the paint in a right cylindrical can is provided by V=4t2−t, where t represents the time in seconds and V represents the volume in the cubic centimeters.

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Chapter 12.5, Problem 44E

Chapter 12.5, Problem 46E

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Chapter 12 Solutions

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Solution Summary: The author calculates the rate at which the level rising is 1.6cm/sec.

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