I made a problem to challenge my friends and now I am puzzled myself how to solve it. I have came up with a solution, but I want to cross-check with someone. So here's the question:
There is a cake with a height of 1 in and a diameter of 2 in. A slice of the cake has been cut, and the curved side of the slice of cake is 1 in. How much cream (surface area) is left on the cake after the slice has been taken away from the cake? (Keep in mind the instep of the slice doesn't leave any cream, and the bottom of the cake doesn't need cream)
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My solution:
(full cake top area – slice top area) = consumed cake's top area
(full cake side area – slice side area) = consumed cake's side area
consumed cake's top area + consumed cake's side area = consumed cake's cream.
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full cake top area = (pi * r^2) = pi * 1^2 = 1pi or pi
full cake side area = circumference * height = 2pi * 1 = 2pi
slice fraction of cake = slice side length / circumference of cake = 1/2pi
slice top area = full cake top area * (slice fraction of cake) = pi * (1/2pi) = 1/2
slice side area = slice side * height = 1 * 1 = 1
consumed cake's top area = pi – 1/2
consumed cake's side area = 2pi – 1
consumed cake's cream area = consumed cake's top area + consumed cake's side area = (pi – 1/2) + (2pi – 1) = 3pi – 1.5 = approx. 7.92477796 or 8 in^2
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Very sorry if the math didn't make since, let me know for any clarifications.
Best Answer
The math all looks just fine! It's odd to think of a cake that small, but the math definitely checks out. The exact answer is, indeed, $3\pi-\frac32$ square inches of creamed surface area.