Maths

[Tam]

Thie is probably the wrong section, but I need to know the surface area of a cup with the following dimensions: Height - 8.0cm Diameter of base - 5.0cm (filled) Diameter of top - 7.0cm (open)

[LeMarchand]

Part1: It's been years since I've done geometry (so there's probably a much easier way of doing this, and I'm not claiming that I've got this 100% correct, plus it would be a lot easier to draw a diagram!), but my suggestion would be to extrapolate the cup into a cone (imagine inverting it and drawing along the outside edge), then work out the surface area of the whole cone. If you subtract the surface area of the "extra" cone on top, you will have the surface area of your cup excluding the base, which you can use �r� to resolve. To find the area of a cone, you use �r�+�rs (base area + curved surface area), where 's' is the sloping side. We only need �rs, as the 'base' of the cone is the open top.

[LeMarchand]

Part 2: So you have your inverted cup, with the base of your imaginary cone the open top, so r=7/2=3.5. Imagine the cup as a triangle either side, and two rectangles in the middle. The height of the rectangles is the height of the cup (8cm) and the width is �� the width of the base of the cup (5/2=2.5) therefore the triangles must be 1cm at the base and 8cm high. Using good old Pythagoras, the square of the other side of the triangle must be 1��+8��=65, so its length is 8.06cm. What about the length of the slope on the ��extra�� cone? Imagine it��s two triangles, we need to know two sides or a side and one angle. Well, one side is �� the base of the cup (5/2=2.5), and the angle at the top must be the same as the angle as the top of the ��lower�� triangles (180�� in any triangle, and we have 90�� from the rectangle and the angle at the top of the lower triangle). Use trig to get this (7.13�� - the tangent of the angle is 1/8, and you can then find the inverse of this to work out the angle), or cheat and use http://www.1728.com/pythgorn.htm . We can use the same methods (to cheat, use http://www.1728.com/trig.htm ) to determine the sloping side of the top triangle, getting 20.14. So the area of the whole cone is ��rs=3.14*3.5*(20.14+8.06)=309.92cm��. The area of the top cane is 3.14*2.5*20.14=158.10cm��, so the area of the cup excluding its base is 309.92-158.10=151.82cm��. The area of the base is ��r��=3.14*2.5*2.5=19.63, so the total area of the cup is 151.82+19.63=171.45 cm��. Probably too late for your homework, though. P.S. the funny squiggle was supposed to be "pi".

[LeMarchand]

P.P.S., the copyright symbol followed by "divided by" was the "square" sign on my PC, and ignore any "Yen" symbols. It all looked OK on my screen, honest! If only we could preview our replies!

[sddsddean]

To help summarise LeMarchand's answer, go to http://www.roymech.co.uk/Useful_Tables/Form/Mathem
atics.html and you want the area of frustrum of cone. (A cup is an upside down version), then take away the area of a circle for the top (7cm dia) to give you the area of the cup.

[sddsddean]

Forgot to say, l(length of the side) = 8.0622577cm (by Pythagoras) as LeMarchand said.