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An open box of maximum volume is to be made from a square piece of material, s = 36 centimeters on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Height, x Length and Width Volume, V 1 36 − 2(1) 1[36 − 2(1)]2 = 1156 2 36 − 2(2) 2[36 − 2(2)]2 = 2048 3 36 − 2(3) 3[36 − 2(3)]2 =  4 36 − 2(4) 4[36 − 2(4)]2 =  5 36 − 2(5) 5[36 − 2(5)]2 =  6 36 − 2(6) 6[36 − 2(6)]2 =  Use the table to guess the maximum volume. V =  (b) Write the volume V as a function of x. V =        ,      0 < x < 18 (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. V =  (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

An open box of maximum volume is to be made from a square piece of material, s = 36 centimeters on a side, by cutting equal squares from the corners and turning up the sides (see figure). (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) Height, x Length and Width Volume, V 1 36 − 2(1) 1[36 − 2(1)]2 = 1156 2 36 − 2(2) 2[36 − 2(2)]2 = 2048 3 36 − 2(3) 3[36 − 2(3)]2 =  4 36 − 2(4) 4[36 − 2(4)]2 =  5 36 − 2(5) 5[36 − 2(5)]2 =  6 36 − 2(6) 6[36 − 2(6)]2 =  Use the table to guess the maximum volume. V =  (b) Write the volume V as a function of x. V =        ,      0 < x < 18 (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. V =  (d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.6: Variation
Problem 9E
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An open box of maximum volume is to be made from a square piece of material, s = 36 centimeters on a side, by cutting equal squares from the corners and turning up the sides (see figure).

(a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.)

Height, x Length and
Width		 Volume, V
1 36 − 2(1) 1[36 − 2(1)]2 = 1156
2 36 − 2(2) 2[36 − 2(2)]2 = 2048
3 36 − 2(3) 3[36 − 2(3)]2 = 
4 36 − 2(4) 4[36 − 2(4)]2 = 
5 36 − 2(5) 5[36 − 2(5)]2 = 
6 36 − 2(6) 6[36 − 2(6)]2 = 


Use the table to guess the maximum volume.
V = 

(b) Write the volume V as a function of x.

V =
 
 
 
 ,		      0 < x < 18



(c) Use calculus to find the critical number of the function in part (b) and find the maximum value.
V = 

(d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph.

An open box of maximum volume is to be made from a square piece of material, s = 36 centimeters on a side, by cutting equal squares from the corners and turning up the sides (see figure). s- 2x (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) |Height, x Length and Width Volume, V 1 36 – 2(1) 1[36 – 2(1)]2 = 1156 2 36 – 2(2) 2[36 – 2(2)]2 = 2048 3 36 – 2(3) 3[36 – 2(3)]2 = 4 36 – 2(4) 4[36 – 2(4)]2 = 5 36 – 2(5) 5[36 – 2(5)]2 6 36 – 2(6) 6[36 – 2(6)]² = Use the table to guess the maximum volume. V = (b) Write the volume V as a function of x. V = 0 < x < 18 (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. V= – 2x
Transcribed Image Text:An open box of maximum volume is to be made from a square piece of material, s = 36 centimeters on a side, by cutting equal squares from the corners and turning up the sides (see figure). s- 2x (a) Analytically complete six rows of a table such as the one below. (The first two rows are shown.) |Height, x Length and Width Volume, V 1 36 – 2(1) 1[36 – 2(1)]2 = 1156 2 36 – 2(2) 2[36 – 2(2)]2 = 2048 3 36 – 2(3) 3[36 – 2(3)]2 = 4 36 – 2(4) 4[36 – 2(4)]2 = 5 36 – 2(5) 5[36 – 2(5)]2 6 36 – 2(6) 6[36 – 2(6)]² = Use the table to guess the maximum volume. V = (b) Write the volume V as a function of x. V = 0 < x < 18 (c) Use calculus to find the critical number of the function in part (b) and find the maximum value. V= – 2x
(d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph. V V 3500 3000 1500 2500 2000 1000 1500 1000 500- 500 X 10 15 5 10 15 V V 3500 400 3000 2500 300 2000 200 1500 1000 100 500 6 8 5 10 15
Transcribed Image Text:(d) Use a graphing utility to graph the function in part (b) and verify the maximum volume from the graph. V V 3500 3000 1500 2500 2000 1000 1500 1000 500- 500 X 10 15 5 10 15 V V 3500 400 3000 2500 300 2000 200 1500 1000 100 500 6 8 5 10 15
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