Question 1105171: please solve
a graduated cylinder, pictured, is used to measure solutions in a laboratory. It is open at the top and closed at the bottom.
If a graduated cylinder has a height (h) that is 4 cm more than its radius (r) and a surface area of 16p cm2, use the equations below to calculate the diameter of the graduated cylinder. Give exact answer in fraction form.
Area of the side of a cylinder = 2 pi r h. Area of the top or bottom of a cylinder = pi r squared
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the lateral surface area of the cylinder is equal to 2 * pi * r * h
the bottom surface area of the cylinder is equal to pi * r^2.
the total surface area of the cylinder is the lateral surface area and the area of the base on the bottom.
you are given that the height of the cylinder is equal to the length of the radius plus 4 cm.
that makes the height of the cylinder equal to r+4.
the total surface area of the cylinder is equal to (2 * pi * r * h) + (pi * r^2).
since h = r+4, then the formula becomes (2 * pi * r * (r+4)) + (pi * r^2)
you are given that the total surface area is equal to 16 * pi.
therefore, 16 * pi = (2 * pi * r * (r+4)) + (pi * r^2)
divide both sides of this equation by pi to get 16 = (2 * r * (r+4)) + r^2
simplify this equation to get 16 = 2 * (r^2 + 4r) + r^2
simplify further to get 16 = 2r^2 + 8r + r^2
combine like terms to get 16 = 3r^2 + 8r
subtract 16 from both sides of this equation to get 0 = 3r^2 + 8r - 16
factor this equation to get (3r-4) * (r+4) = 0
solve for r to get r = 4/3 or r = -4.
r can't be negative, so r = 4/3 should be the value of your radius.
if true, that means the height of the cylinder should be 4 + 4/3 = 12/3 + 4/3 = 16/3.
the radius of the base of the cylinder is 4/3.
the height of the cylinder is 16/3.
the surface area of the cylinder is equal to (2 * pi * r * (r+4)) + (pi * r^2) which becomes (2 * pi * 4/3 * 16/3) + (pi * (4/3)^2) which is equal to 16 * pi.
formula looks good.
diameter of the cylinder is equal to 2 * the radius which is equal to 2 * 4/3 which is equal to 8/3 inches.
if you wish to convert the improper fraction to a mixed fraction, then the radius is equal to 1 and 1/3 inches, and the diameter is equal to 2 and 2/3 inches.
the length of the diameter is your solution.
that is equal to 8/3 inches, or 2 2/3 inches, whichever way you wish to present it.
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