Surface Area of a Cone?

Question

Ok, so ive working on this problem for days and cant figure it out,

A cone shaped drinking cup is to be made to hold 27cm3 of water, find the height and radius of the cup that will use the smallest amout of paper.

If anyone has any ideas i would appreciate it.

Answer 1

Volume = 1/3 * pi * H * R^2
so H= 3 * Volume / (pi * R^2) = 3^4 / (pi * R^2)

SA = pi * R * sqrt(R^2 + H^2) =
pi * R * sqrt(R^2 + (3^4/(pi*R^2))^2) =
pi * sqrt (R^4 + 3^8 / (pi^2 * R^2))

take the derivative of SA with respect to R, set to 0, and solve for R.

[I solved it and got R^6 = 3^8 / (2 * pi^2)) or R = 2.63 approx]

Answer 2

The volume of a cone is given by 1/3 pi r^2h

1/3 pi r^2 h = 27

pi r^2 h = 71 cm^3

r^2 = 22.6 cm^3 / h

Now, surface area of a cone is given by:

pi r * sqrt [r^2+h^2]

Solving and minimizing yields

A height of approximately 3.77, a radius of 2.61 and an area of about 37.68.

Regards,

Mysstere

Answer 3

The formula for finding the area of a right circular cone is:

area = pi * radius (radius + (square root of (radius^2 + height^2))

Answer 4

volume = 1/3(area of base) x (altitude)
volume = 27 cm^3

if the base is a circle ... the area of a circle is 1/4 pi D^2

v=1/3(1/4piD^2) x (altitude)

the units are right ... all I know to do from here is trial and error numbers for diameter, solve for altitude and substitute that number back in to see if it works.

Sorry I can't be more help. Graduated from engr. school in '80 and haven't used this stuff much since then.

Answer 5

Oh.....is it 2/3 x pi r squared h?

It's 40+ years since I learned it, you know.

Answer 6

lateral surface areea of an circular cone=
phi*(a-b)*sqrt(h^2+(+b-a)^2)=
Phi*(a-b)*l

a,b lower and upper Radius
h the high of the cone
l= the lenght of the cone

---------------a-----
l----------xxxxxxxxx-
l---------xxxxxxxxxxx l
l---h---xxxxxxxxxxxxx
l------xxxxxxxxxxxxxxx-
---------------b