Explanation Result used: The Second Derivative Test for Concavity: Let y = f ( x ) be twice differentiable on an interval I . 1. If f ″ > 0 on I , the graph of f over I is concave up. 2. If f ″ < 0 on I , the graph of f over I is concave down. Definition used: Critical point: A critical point of a function f is any number x in the domain of f such that f ′ ( x ) = 0 or f ′ ( x ) = 0 does not exist. Points of Inflection: A point ( c , f ( c ) ) where the graph of a function has a tangent line and where the concavity changes is a point of inflection. Calculation: The width of the box is x . The length and height of the box is given by l = 12 − 2 x and h = 8 − 2 x . Obtain the volume of the box as follows. V = l h w = x ( 8 − 2 x ) ( 12 − 2 x ) = ( 8 x − 2 x 2 ) ( 12 − 2 x ) = 96 x − 24 x 2 − 16 x 2 + 4 x 3 Simplify the above expression to get, V = 4 x 3 − 40 x 2 + 96 x Find the critical points by equating first derivative to zero. 0 = 12 x 2 − 80 x + 96 12 x 2 − 80 x + 96 = 0 4 ( 3 x 2 − 20 x + 24 ) = 0 3 x 2 − 20 x + 24 = 0 Apply quadratic equation formula to get, x 1 , 2 = − b ± b 2 − 4 a c 2 a = − ( − 80 ) ± ( − 80 ) 2 − 4 ⋅ 12 ⋅ 96 2 ⋅ 12 = 2 ( 5 ± 7 ) 3 Simplify the above expression to get, x = 5.09716 , 1.5694 Since 2 x should be lesser than 8 , reject x = 5.09716 . Choose the test value 1 and 2 placed to the left and right of critical point x = 1.6 and choose the test value 2 and 6 placed to the left and right of critical point x = 77.4 . Substitute the test value x = 1 in first derivative and check the sign of the derivative as follows

Basic Technical Mathematics with Calculus (11th Edition)

11th Edition

ISBN: 9780134437736

Author: Allyn J. Washington, Richard Evans

Publisher: PEARSON

See similar textbooks

Question

Chapter 24.5, Problem 53E

To determine

To express: The volume of the box as a function of the side of the box and sketch the graph of the resulting equation.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 24.5, Problem 52E

Chapter 24.5, Problem 54E

Students have asked these similar questions

9. Define a 7-system. Show that P = {(0, x]; (0, 1]} is a л-system.

25. Show that if X is a random variable and g(.) is a Borel measurable function, then Y = g(X) is a random variable.

24. A factory produces items from two machines: Machine A and Machine B. Machine A produces 60% of the total items, while Machine B produces 40%. The probability that an item produced by Machine A is defective is P(D|A)=0.03. The probability that an item produced by Machine B is defective is P(D|B) = 0.05. (a) What is the probability that a randomly selected product be defective, P(D)? (b) If a randomly selected item from the production line is defective, calculate the probability that it was produced by Machine A, P(A|D).

Chapter 24 Solutions

Basic Technical Mathematics with Calculus (11th Edition)

Ch. 24.1 - For the parabola y = 4 − x2, at the point (3, −5)...Ch. 24.1 - Prob. 2PE Ch. 24.1 - Prob. 1E Ch. 24.1 - Prob. 2E Ch. 24.1 - Prob. 3E Ch. 24.1 - Prob. 4E Ch. 24.1 - Prob. 5E Ch. 24.1 - Prob. 6E Ch. 24.1 - Prob. 7E Ch. 24.1 - Prob. 8E

Ch. 24.1 - Prob. 9E Ch. 24.1 - Prob. 10E Ch. 24.1 - Prob. 11E Ch. 24.1 - Prob. 12E Ch. 24.1 - In Exercises 11–14, find the equations of the...Ch. 24.1 - Prob. 14E Ch. 24.1 - Prob. 15E Ch. 24.1 - Prob. 16E Ch. 24.1 - Prob. 17E Ch. 24.1 - Prob. 18E Ch. 24.1 - Prob. 19E Ch. 24.1 - Prob. 20E Ch. 24.1 - Prob. 21E Ch. 24.1 - Where does the normal line to the parabola y = x —...Ch. 24.1 - Prob. 23E Ch. 24.1 - Prob. 24E Ch. 24.1 - A certain suspension cable with supports on the...Ch. 24.1 - Prob. 26E Ch. 24.1 - Prob. 27E Ch. 24.1 - Prob. 28E Ch. 24.1 - Prob. 29E Ch. 24.1 - Prob. 30E Ch. 24.2 - In Example 1, let x1 = 0.3, and find x2. EXAMPLE...Ch. 24.2 - Prob. 1E Ch. 24.2 - Prob. 2E Ch. 24.2 - Prob. 3E Ch. 24.2 - Prob. 4E Ch. 24.2 - Prob. 5E Ch. 24.2 - Prob. 6E Ch. 24.2 - Prob. 7E Ch. 24.2 - Prob. 8E Ch. 24.2 - Prob. 9E Ch. 24.2 - Prob. 10E Ch. 24.2 - Prob. 11E Ch. 24.2 - Prob. 12E Ch. 24.2 - Prob. 13E Ch. 24.2 - Prob. 14E Ch. 24.2 - Prob. 15E Ch. 24.2 - Prob. 16E Ch. 24.2 - Prob. 17E Ch. 24.2 - Prob. 18E Ch. 24.2 - Prob. 19E Ch. 24.2 - Prob. 20E Ch. 24.2 - Prob. 21E Ch. 24.2 - Prob. 23E Ch. 24.2 - Prob. 24E Ch. 24.2 - Prob. 25E Ch. 24.2 - Prob. 27E Ch. 24.2 - Prob. 28E Ch. 24.2 - Prob. 29E Ch. 24.2 - Prob. 30E Ch. 24.3 - Prob. 1PE Ch. 24.3 - Prob. 1E Ch. 24.3 - Prob. 2E Ch. 24.3 - Prob. 3E Ch. 24.3 - Prob. 4E Ch. 24.3 - Prob. 5E Ch. 24.3 - Prob. 6E Ch. 24.3 - Prob. 7E Ch. 24.3 - Prob. 8E Ch. 24.3 - Prob. 9E Ch. 24.3 - Prob. 10E Ch. 24.3 - Prob. 11E Ch. 24.3 - Prob. 12E Ch. 24.3 - Prob. 13E Ch. 24.3 - Prob. 14E Ch. 24.3 - Prob. 15E Ch. 24.3 - Prob. 16E Ch. 24.3 - In Exercises 11–30, find the indicated velocities...Ch. 24.3 - Prob. 18E Ch. 24.3 - Prob. 19E Ch. 24.3 - Prob. 20E Ch. 24.3 - Prob. 21E Ch. 24.3 - Prob. 22E Ch. 24.3 - Prob. 23E Ch. 24.3 - Prob. 24E Ch. 24.3 - Prob. 25E Ch. 24.3 - Prob. 26E Ch. 24.3 - Prob. 27E Ch. 24.3 - Prob. 28E Ch. 24.3 - Prob. 29E Ch. 24.3 - Prob. 30E Ch. 24.4 - In Example 2, change each 10 to 12, and then...Ch. 24.4 - In Exercises 1 and 2, make the given changes in...Ch. 24.4 - In Exercises 1 and 2, make the given changes in...Ch. 24.4 - In Exercises 3–6, assume that all variables are...Ch. 24.4 - In Exercises 3–6, assume that all variables are...Ch. 24.4 - In Exercises 3–6, assume that all variables are...Ch. 24.4 - In Exercises 3–6, assume that all variables are...Ch. 24.4 - Prob. 7E Ch. 24.4 - Prob. 8E Ch. 24.4 - Prob. 9E Ch. 24.4 - Prob. 10E Ch. 24.4 - Prob. 11E Ch. 24.4 - Prob. 12E Ch. 24.4 - Prob. 13E Ch. 24.4 - Prob. 14E Ch. 24.4 - Prob. 15E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 17E Ch. 24.4 - Prob. 18E Ch. 24.4 - Prob. 19E Ch. 24.4 - Prob. 20E Ch. 24.4 - Prob. 21E Ch. 24.4 - Prob. 22E Ch. 24.4 - Prob. 23E Ch. 24.4 - Prob. 24E Ch. 24.4 - Prob. 25E Ch. 24.4 - Prob. 26E Ch. 24.4 - Prob. 27E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 31E Ch. 24.4 - Prob. 32E Ch. 24.4 - Prob. 33E Ch. 24.4 - Prob. 34E Ch. 24.4 - Prob. 35E Ch. 24.4 - Prob. 36E Ch. 24.4 - In Exercises 7–42, solve the problems in related...Ch. 24.4 - Prob. 38E Ch. 24.4 - Prob. 39E Ch. 24.4 - Prob. 40E Ch. 24.4 - Prob. 41E Ch. 24.4 - Prob. 42E Ch. 24.5 - Prob. 1PE Ch. 24.5 - Prob. 2PE Ch. 24.5 - Prob. 1E Ch. 24.5 - Prob. 2E Ch. 24.5 - Prob. 3E Ch. 24.5 - Prob. 4E Ch. 24.5 - Prob. 5E Ch. 24.5 - Prob. 6E Ch. 24.5 - Prob. 7E Ch. 24.5 - Prob. 8E Ch. 24.5 - Prob. 9E Ch. 24.5 - Prob. 10E Ch. 24.5 - Prob. 11E Ch. 24.5 - Prob. 12E Ch. 24.5 - Prob. 13E Ch. 24.5 - Prob. 14E Ch. 24.5 - Prob. 15E Ch. 24.5 - Prob. 16E Ch. 24.5 - Prob. 17E Ch. 24.5 - Prob. 18E Ch. 24.5 - Prob. 19E Ch. 24.5 - Prob. 20E Ch. 24.5 - Prob. 21E Ch. 24.5 - Prob. 22E Ch. 24.5 - Prob. 23E Ch. 24.5 - Prob. 24E Ch. 24.5 - Prob. 25E Ch. 24.5 - Prob. 26E Ch. 24.5 - Prob. 27E Ch. 24.5 - Prob. 28E Ch. 24.5 - Prob. 29E Ch. 24.5 - Prob. 30E Ch. 24.5 - Prob. 31E Ch. 24.5 - Prob. 32E Ch. 24.5 - Prob. 33E Ch. 24.5 - Prob. 34E Ch. 24.5 - Prob. 35E Ch. 24.5 - Prob. 36E Ch. 24.5 - Prob. 37E Ch. 24.5 - Prob. 38E Ch. 24.5 - Prob. 39E Ch. 24.5 - Prob. 40E Ch. 24.5 - Prob. 41E Ch. 24.5 - Prob. 42E Ch. 24.5 - Prob. 43E Ch. 24.5 - Prob. 44E Ch. 24.5 - Prob. 45E Ch. 24.5 - Prob. 46E Ch. 24.5 - Prob. 47E Ch. 24.5 - Prob. 48E Ch. 24.5 - Prob. 49E Ch. 24.5 - Prob. 50E Ch. 24.5 - Prob. 51E Ch. 24.5 - Prob. 52E Ch. 24.5 - Prob. 53E Ch. 24.5 - Prob. 54E Ch. 24.5 - Prob. 55E Ch. 24.5 - Prob. 56E Ch. 24.5 - Prob. 57E Ch. 24.5 - Prob. 58E Ch. 24.6 - Prob. 1PE Ch. 24.6 - Prob. 1E Ch. 24.6 - Prob. 2E Ch. 24.6 - Prob. 3E Ch. 24.6 - Prob. 4E Ch. 24.6 - Prob. 5E Ch. 24.6 - Prob. 6E Ch. 24.6 - Prob. 7E Ch. 24.6 - Prob. 8E Ch. 24.6 - Prob. 9E Ch. 24.6 - Prob. 10E Ch. 24.6 - Prob. 11E Ch. 24.6 - Prob. 12E Ch. 24.6 - Prob. 13E Ch. 24.6 - Prob. 14E Ch. 24.6 - Prob. 15E Ch. 24.6 - Prob. 16E Ch. 24.6 - Prob. 17E Ch. 24.6 - Prob. 18E Ch. 24.6 - Prob. 19E Ch. 24.6 - Prob. 20E Ch. 24.6 - Prob. 21E Ch. 24.6 - Prob. 22E Ch. 24.6 - Prob. 23E Ch. 24.6 - Prob. 24E Ch. 24.6 - Prob. 25E Ch. 24.6 - Prob. 26E Ch. 24.6 - Prob. 27E Ch. 24.6 - Prob. 28E Ch. 24.6 - Prob. 29E Ch. 24.6 - Prob. 30E Ch. 24.6 - Prob. 31E Ch. 24.6 - Prob. 32E Ch. 24.7 - Prob. 1PE Ch. 24.7 - Prob. 2PE Ch. 24.7 - Prob. 1E Ch. 24.7 - Prob. 2E Ch. 24.7 - The height (in ft) of a flare shot upward from the...Ch. 24.7 - Prob. 4E Ch. 24.7 - Prob. 5E Ch. 24.7 - Prob. 6E Ch. 24.7 - Prob. 7E Ch. 24.7 - Prob. 8E Ch. 24.7 - Prob. 9E Ch. 24.7 - Prob. 10E Ch. 24.7 - Prob. 11E Ch. 24.7 - Prob. 12E Ch. 24.7 - In deep water, the velocity of a wave is , where a...Ch. 24.7 - Prob. 14E Ch. 24.7 - Prob. 15E Ch. 24.7 - Prob. 16E Ch. 24.7 - A microprocessor chip is being designed with a...Ch. 24.7 - Prob. 18E Ch. 24.7 - What are the dimensions of the largest rectangular...Ch. 24.7 - A rectangular storage area is to be constructed...Ch. 24.7 - Prob. 21E Ch. 24.7 - Prob. 22E Ch. 24.7 - Prob. 23E Ch. 24.7 - Prob. 24E Ch. 24.7 - Prob. 25E Ch. 24.7 - Prob. 26E Ch. 24.7 - Prob. 27E Ch. 24.7 - Prob. 28E Ch. 24.7 - Prob. 29E Ch. 24.7 - Prob. 30E Ch. 24.7 - Prob. 31E Ch. 24.7 - Prob. 32E Ch. 24.7 - Prob. 33E Ch. 24.7 - What is the minimum slope of the curve y = x5 −...Ch. 24.7 - Prob. 35E Ch. 24.7 - Prob. 36E Ch. 24.7 - Prob. 37E Ch. 24.7 - Prob. 38E Ch. 24.7 - Prob. 39E Ch. 24.7 - Prob. 40E Ch. 24.7 - Prob. 41E Ch. 24.7 - Computer simulation shows that the drag F (in N)...Ch. 24.7 - Prob. 43E Ch. 24.7 - The potential energy E of an electric charge q due...Ch. 24.7 - An open box is to be made from a square piece of...Ch. 24.7 - Prob. 46E Ch. 24.7 - Prob. 47E Ch. 24.7 - Prob. 48E Ch. 24.7 - An oil pipeline is to be built from a refinery to...Ch. 24.7 - Prob. 50E Ch. 24.7 - Prob. 51E Ch. 24.7 - Prob. 52E Ch. 24.7 - Prob. 53E Ch. 24.7 - Prob. 54E Ch. 24.8 - Prob. 1PE Ch. 24.8 - Prob. 2PE Ch. 24.8 - Prob. 1E Ch. 24.8 - Prob. 2E Ch. 24.8 - Prob. 3E Ch. 24.8 - Prob. 4E Ch. 24.8 - Prob. 5E Ch. 24.8 - Prob. 6E Ch. 24.8 - Prob. 7E Ch. 24.8 - Prob. 8E Ch. 24.8 - Prob. 9E Ch. 24.8 - Prob. 10E Ch. 24.8 - Prob. 11E Ch. 24.8 - Prob. 12E Ch. 24.8 - Prob. 13E Ch. 24.8 - Prob. 14E Ch. 24.8 - Prob. 15E Ch. 24.8 - Prob. 16E Ch. 24.8 - Prob. 17E Ch. 24.8 - Prob. 18E Ch. 24.8 - Prob. 19E Ch. 24.8 - Prob. 20E Ch. 24.8 - Prob. 21E Ch. 24.8 - Prob. 22E Ch. 24.8 - Prob. 23E Ch. 24.8 - Prob. 24E Ch. 24.8 - Prob. 25E Ch. 24.8 - Prob. 26E Ch. 24.8 - Prob. 27E Ch. 24.8 - Prob. 28E Ch. 24.8 - Prob. 29E Ch. 24.8 - Prob. 30E Ch. 24.8 - Prob. 31E Ch. 24.8 - Prob. 32E Ch. 24.8 - Prob. 33E Ch. 24.8 - Prob. 34E Ch. 24.8 - Prob. 35E Ch. 24.8 - Prob. 36E Ch. 24.8 - Prob. 37E Ch. 24.8 - Prob. 38E Ch. 24.8 - Prob. 39E Ch. 24.8 - Prob. 40E Ch. 24.8 - Prob. 41E Ch. 24.8 - Prob. 42E Ch. 24.8 - Prob. 43E Ch. 24.8 - Prob. 44E Ch. 24 - Prob. 1RE Ch. 24 - Prob. 2RE Ch. 24 - Prob. 3RE Ch. 24 - Prob. 4RE Ch. 24 - Prob. 5RE Ch. 24 - Prob. 6RE Ch. 24 - Prob. 7RE Ch. 24 - Prob. 8RE Ch. 24 - Prob. 9RE Ch. 24 - Prob. 10RE Ch. 24 - Prob. 11RE Ch. 24 - Prob. 12RE Ch. 24 - Prob. 13RE Ch. 24 - Prob. 14RE Ch. 24 - Prob. 15RE Ch. 24 - Prob. 16RE Ch. 24 - Prob. 17RE Ch. 24 - Prob. 18RE Ch. 24 - Prob. 19RE Ch. 24 - Prob. 20RE Ch. 24 - Prob. 21RE Ch. 24 - Prob. 22RE Ch. 24 - Prob. 23RE Ch. 24 - Prob. 24RE Ch. 24 - Prob. 25RE Ch. 24 - Prob. 26RE Ch. 24 - Prob. 27RE Ch. 24 - In Exercises 25–32, sketch the graphs of the given...Ch. 24 - Prob. 29RE Ch. 24 - Prob. 30RE Ch. 24 - Prob. 31RE Ch. 24 - Prob. 32RE Ch. 24 - Prob. 33RE Ch. 24 - Prob. 34RE Ch. 24 - Prob. 35RE Ch. 24 - Prob. 36RE Ch. 24 - Prob. 37RE Ch. 24 - Prob. 38RE Ch. 24 - Prob. 39RE Ch. 24 - Prob. 40RE Ch. 24 - Prob. 41RE Ch. 24 - Prob. 42RE Ch. 24 - Prob. 43RE Ch. 24 - Prob. 44RE Ch. 24 - Prob. 45RE Ch. 24 - Prob. 46RE Ch. 24 - Prob. 47RE Ch. 24 - Prob. 48RE Ch. 24 - Prob. 49RE Ch. 24 - Prob. 50RE Ch. 24 - Prob. 51RE Ch. 24 - Prob. 52RE Ch. 24 - In Exercises 49–94, solve the given problems. 53....Ch. 24 - Prob. 54RE Ch. 24 - Prob. 55RE Ch. 24 - Prob. 56RE Ch. 24 - The deflection y (in m) of a beam at a horizontal...Ch. 24 - Prob. 58RE Ch. 24 - Prob. 59RE Ch. 24 - Prob. 60RE Ch. 24 - Prob. 61RE Ch. 24 - Prob. 62RE Ch. 24 - In Fig. 24.75, the tension T supports the 40.0-N...Ch. 24 - Prob. 64RE Ch. 24 - Prob. 65RE Ch. 24 - Prob. 66RE Ch. 24 - An analysis of the power output P (in kW/m3) of a...Ch. 24 - The altitude h (in ft) of a certain rocket as a...Ch. 24 - Prob. 69RE Ch. 24 - Prob. 70RE Ch. 24 - Prob. 71RE Ch. 24 - Prob. 72RE Ch. 24 - Prob. 73RE Ch. 24 - A special insulation strip is to be sealed...Ch. 24 - Prob. 75RE Ch. 24 - Prob. 76RE Ch. 24 - Prob. 77RE Ch. 24 - Prob. 78RE Ch. 24 - Prob. 79RE Ch. 24 - Prob. 80RE Ch. 24 - Prob. 81RE Ch. 24 - Prob. 82RE Ch. 24 - Prob. 83RE Ch. 24 - Prob. 84RE Ch. 24 - Prob. 85RE Ch. 24 - Prob. 86RE Ch. 24 - Prob. 87RE Ch. 24 - Prob. 88RE Ch. 24 - Prob. 89RE Ch. 24 - Prob. 90RE Ch. 24 - Prob. 91RE Ch. 24 - Prob. 92RE Ch. 24 - Prob. 93RE Ch. 24 - Prob. 94RE Ch. 24 - Prob. 95RE Ch. 24 - Prob. 1PT Ch. 24 - Prob. 2PT Ch. 24 - Prob. 3PT Ch. 24 - Prob. 4PT Ch. 24 - Prob. 5PT Ch. 24 - Prob. 6PT Ch. 24 - Prob. 7PT Ch. 24 - Prob. 8PT Ch. 24 - Prob. 9PT Ch. 24 - Prob. 10PT

Knowledge Booster