The table below gives the approximate measurement in inches of some common items that are packed in cylindrical containers. Radius (inches) Volume (cubic inches) Product Height (inches) Baking power 1.25 3.65 17.92 1.45 7.50 Cleanser 49.54 Coffee 1.95 5.20 62.12 Frosting 1.63 3.60 30.05 2.10 6.70 Pineapple juice 92.82 7. A. Begin with the baking powder. The volume of the baking powder can be expressed as an equation Find the equation for the volume of a cylinder (or work it out yourself. area of base x height). Set this equation equal to the volume of the baking powder and solve for the height h with respect to the variable r. B. Next, write out the surface area formula. This should contain two independent variables, r and h. You can constrain this formula to the given volume by substituting the h with the expression of h(r) you found in step 8. You should now have an equation containing only the variable r. Call this equation S(r) for surface area dependent upon radius. Without technology find the derivative of your surface area function S'(7). C. D. Graph your derivative in the same viewing window as your surface area function and look at the graph. (Set yourplot range to show portions that apply to this situation, i.e. no negative radius.) Make sure the window settings on your graph identifies a distinct minimum surface area. Minimum values will occur at x-values that make the derivative equal to 0 or undefined. Mark these coordinates on your graph. E Without technology solve for minimized value ofr F. Use the value for r from step 5 to determine the minimized height and surface area for the baking powder. (Show all work!)
The table below gives the approximate measurement in inches of some common items that are packed in cylindrical containers. Radius (inches) Volume (cubic inches) Product Height (inches) Baking power 1.25 3.65 17.92 1.45 7.50 Cleanser 49.54 Coffee 1.95 5.20 62.12 Frosting 1.63 3.60 30.05 2.10 6.70 Pineapple juice 92.82 7. A. Begin with the baking powder. The volume of the baking powder can be expressed as an equation Find the equation for the volume of a cylinder (or work it out yourself. area of base x height). Set this equation equal to the volume of the baking powder and solve for the height h with respect to the variable r. B. Next, write out the surface area formula. This should contain two independent variables, r and h. You can constrain this formula to the given volume by substituting the h with the expression of h(r) you found in step 8. You should now have an equation containing only the variable r. Call this equation S(r) for surface area dependent upon radius. Without technology find the derivative of your surface area function S'(7). C. D. Graph your derivative in the same viewing window as your surface area function and look at the graph. (Set yourplot range to show portions that apply to this situation, i.e. no negative radius.) Make sure the window settings on your graph identifies a distinct minimum surface area. Minimum values will occur at x-values that make the derivative equal to 0 or undefined. Mark these coordinates on your graph. E Without technology solve for minimized value ofr F. Use the value for r from step 5 to determine the minimized height and surface area for the baking powder. (Show all work!)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:The table below gives the approximate measurement in inches of some common items that are
packed in cylindrical containers.
Radius (inches)
Volume (cubic inches)
Product
Height (inches)
Baking power
1.25
3.65
17.92
1.45
7.50
Cleanser
49.54
Coffee
1.95
5.20
62.12
Frosting
1.63
3.60
30.05
2.10
6.70
Pineapple juice
92.82
7. A. Begin with the baking powder. The volume of the baking powder can be expressed as an
equation
Find the equation for the volume of a cylinder (or work it out yourself. area of
base x height). Set this equation equal to the volume of the baking powder and solve for the
height h with respect to the variable r.
B. Next, write out the surface area formula. This should contain two independent variables, r
and h. You can constrain this formula to the given volume by substituting the h with the
expression of h(r) you found in step 8. You should now have an equation containing only
the variable r. Call this equation S(r) for surface area dependent upon radius.
Without technology find the derivative of your surface area function S'(7).
C.
D. Graph your derivative in the same viewing window as your surface area function and
look at the graph. (Set yourplot range to show portions that apply to this situation, i.e.
no negative radius.) Make sure the window settings on your graph identifies a distinct
minimum surface area. Minimum values will occur at x-values that make the derivative
equal to 0 or undefined. Mark these coordinates on your graph.
E
Without technology solve for minimized value ofr
F. Use the value for r from step 5 to determine the minimized height and surface area for the
baking powder. (Show all work!)
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