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We'll be analyzing the surface area of a round cylinder - in other words, the amount of material needed to "make a can". A cylinder (round can) has a circular base and a circular top with vertical sides in between. Let r be the radius of the top of the can and let be the height. The surface area of the cylinder, A, is A 2πr² + 2πrh (two circles, one for the top and one for the bottom plus a rolled up rectangle for the sides). = Areas = 1² r = radius Circumference 2лr r (A) = h = height Area = h(2πr) Part a: Assume that the height of your cylinder is 4 inches. Consider A as a function of r, so we can write that as A (r) = 2 r² + 8 πr. What is the domain of A (r)? In other words, for which values of r is A (r) defined? Part b: Continue to assume that the height of your cylinder is 4 inches. Write the radius r as a function of A. This is the inverse function to A (r), i.e., to turn A as a function of r into r as a function of A. Hints: • To calculate an inverse function, you need to solve for r. Here, you would start with A = 2 ² + 8 πr. This equation is the same as 277² +8 πr - A=0 which is a quadratic equation in the variable r, and you can solve that using the quadratic formula. You will want to keep A as a variable when you plug the values into the quadratic formula. • If you want to type in 3 π+1 in Mobius, in text mode you can type in (3*pi+1)/(x+1). There x+1 is more information in the Introduction to Mobius unit. Part c: If the surface area is 150 square inches, then what is the radius r? In other words, evaluate r (150). Round your answer to 2 decimal places. Hint: To compute a numeric square root such as √17.3, you could • Use a spreadsheet such as Microsoft Excel or OpenOffice Calc and type in = sqrt(17.3) • Use a browser to connect to the Internet and type in sqrt(17.3) into a search field • Use a calculator The radius is Number inches if the surface area is 150 square inches.

We'll be analyzing the surface area of a round cylinder - in other words, the amount of material needed to "make a can". A cylinder (round can) has a circular base and a circular top with vertical sides in between. Let r be the radius of the top of the can and let be the height. The surface area of the cylinder, A, is A 2πr² + 2πrh (two circles, one for the top and one for the bottom plus a rolled up rectangle for the sides). = Areas = 1² r = radius Circumference 2лr r (A) = h = height Area = h(2πr) Part a: Assume that the height of your cylinder is 4 inches. Consider A as a function of r, so we can write that as A (r) = 2 r² + 8 πr. What is the domain of A (r)? In other words, for which values of r is A (r) defined? Part b: Continue to assume that the height of your cylinder is 4 inches. Write the radius r as a function of A. This is the inverse function to A (r), i.e., to turn A as a function of r into r as a function of A. Hints: • To calculate an inverse function, you need to solve for r. Here, you would start with A = 2 ² + 8 πr. This equation is the same as 277² +8 πr - A=0 which is a quadratic equation in the variable r, and you can solve that using the quadratic formula. You will want to keep A as a variable when you plug the values into the quadratic formula. • If you want to type in 3 π+1 in Mobius, in text mode you can type in (3*pi+1)/(x+1). There x+1 is more information in the Introduction to Mobius unit. Part c: If the surface area is 150 square inches, then what is the radius r? In other words, evaluate r (150). Round your answer to 2 decimal places. Hint: To compute a numeric square root such as √17.3, you could • Use a spreadsheet such as Microsoft Excel or OpenOffice Calc and type in = sqrt(17.3) • Use a browser to connect to the Internet and type in sqrt(17.3) into a search field • Use a calculator The radius is Number inches if the surface area is 150 square inches.

Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
ChapterA: Appendix
SectionA.2: Geometric Constructions
Problem 9P: A soda can is made from 40 square inches of aluminum. Let x denote the radius of the top of the can,...
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The height of the cylinder is 4 inches. We'll be analyzing the surface area of a round cylinder - in other words, the amount of material needed to "make a can". A cylinder (round can) has a circular base and a circular top with vertical sides in between. Let r be the radius of the top of the can and let h be the height. The surface area of the cylinder, A, is A 2πr² + 2πrh (two circles, one for the top and one for the bottom plus a rolled up rectangle for the sides). = Areas = ₁² r = radius Circumference 2лr r (A) = h = height Area = h(2+) Part a: Assume that the height of your cylinder is 4 inches. Consider A as a function of r, so we can write that as A (r) = 2 πr² + 8 πr. What is the domain of A (r)? In other words, for which values of r is A (r) defined? Part b: Continue to assume that the height of your cylinder is 4 inches. Write the radius r as a function of A. This is the inverse function to A (r), i.e., to turn A as a function of r into r as a function of A. Hints: • To calculate an inverse function, you need to solve for r. Here, you would start with A = 2 r² + 8 πr. This equation is the same as 2 r² +8 πr - A=0 which is a quadratic equation in the variable r, and you can solve that using the quadratic formula. You will want to keep A as a variable when you plug the values into the quadratic formula. • If you want to type in 3 π+1 in Mobius, in text mode you can type in (3*pi+1)/(x+1). There x+1 is more information in the Introduction to Mobius unit. Part c: If the surface area is 150 square inches, then what is the radius r? In other words, evaluate r (150). Round your answer to 2 decimal places. Hint: To compute a numeric square root such as √17.3, you could • Use a spreadsheet such as Microsoft Excel or OpenOffice Calc and type in =sqrt(17.3) • Use a browser to connect to the Internet and type in sqrt(17.3) into a search field • Use a calculator The radius is Number inches if the surface area is 150 square inches.
Transcribed Image Text:The height of the cylinder is 4 inches. We'll be analyzing the surface area of a round cylinder - in other words, the amount of material needed to "make a can". A cylinder (round can) has a circular base and a circular top with vertical sides in between. Let r be the radius of the top of the can and let h be the height. The surface area of the cylinder, A, is A 2πr² + 2πrh (two circles, one for the top and one for the bottom plus a rolled up rectangle for the sides). = Areas = ₁² r = radius Circumference 2лr r (A) = h = height Area = h(2+) Part a: Assume that the height of your cylinder is 4 inches. Consider A as a function of r, so we can write that as A (r) = 2 πr² + 8 πr. What is the domain of A (r)? In other words, for which values of r is A (r) defined? Part b: Continue to assume that the height of your cylinder is 4 inches. Write the radius r as a function of A. This is the inverse function to A (r), i.e., to turn A as a function of r into r as a function of A. Hints: • To calculate an inverse function, you need to solve for r. Here, you would start with A = 2 r² + 8 πr. This equation is the same as 2 r² +8 πr - A=0 which is a quadratic equation in the variable r, and you can solve that using the quadratic formula. You will want to keep A as a variable when you plug the values into the quadratic formula. • If you want to type in 3 π+1 in Mobius, in text mode you can type in (3*pi+1)/(x+1). There x+1 is more information in the Introduction to Mobius unit. Part c: If the surface area is 150 square inches, then what is the radius r? In other words, evaluate r (150). Round your answer to 2 decimal places. Hint: To compute a numeric square root such as √17.3, you could • Use a spreadsheet such as Microsoft Excel or OpenOffice Calc and type in =sqrt(17.3) • Use a browser to connect to the Internet and type in sqrt(17.3) into a search field • Use a calculator The radius is Number inches if the surface area is 150 square inches.
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