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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Equations and inequalities describe the relationship between two mathematical expressions.
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A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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Need help with D, E, and F.
Anser A- h= 17.92/(pi)r^2
Answer B-S(r)= 35.84/r + (pi)r^2
Answer C- S'(r)= -35.84/r^2+2(pi)r
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