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 (it's two circles for the top and bottom plus a rolled up rectangle for the side). 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)=2πr2+8πr. What is the domain of A(r)? In other words, for which values of r is A(r) defined? Part b: Find the inverse function to A(r). Your answer should look like r="some expression involving A". r(A)= Hints: To calculate an inverse function, you need to solve for r. Here you could start with A=2πr2+8πr. This equation is the same as 2πr2+8πr−A=0. Do you recognize this as a quadratic equation ax2+bx+c=0 where the variable x is r? The coefficients would be 2π for a, 8π for b, and −A for c. You can solve for r using the quadratic formula even though the constant term c is a symbol here. Part c: If the surface area is 275 square inches, then what is the radius r? In other words, evaluate r(275). 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 inches if the surface area is 275 square inches.
Cylinders
A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.
Cones
A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.
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 (it's two circles for the top and bottom plus a rolled up rectangle for the side).
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)=2πr2+8πr. What is the domain of A(r)? In other words, for which values of r is A(r) defined?
Part b: Find the inverse function to A(r). Your answer should look like r="some expression involving A".
r(A)=
Hints:
- To calculate an inverse function, you need to solve for r.
- Here you could start with A=2πr2+8πr. This equation is the same as 2πr2+8πr−A=0. Do you recognize this as a
quadratic equation ax2+bx+c=0 where the variable x is r? The coefficients would be 2π for a, 8π for b, and −A for c. - You can solve for r using the quadratic formula even though the constant term c is a symbol here.
Part c: If the surface area is 275 square inches, then what is the radius r? In other words, evaluate r(275). 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 inches if the surface area is 275 square inches.
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