Suppose you are designing a coffee creamer container that has a volume of 48.42 cubic inches. Use the surface area and volume of a cylinder to develope an eqn. relating radius r and surface area S.
¡Ç=pi=3.14
S=2¡Çr^2+2¡Çrh
V=¡Çhr^2
Yes
S = 2 pi r^2 + 2 pi r h
V = pi r^2 h
so
h = 48.42 / pi r^2
so
S = 2 pi r^2 + 2 pi r (48.42 /pi r^2)
S = 2 pi r^2 + 96.84/r
Now I am sure you want to find what r is best for this volume (uses the least metal for area)
dS/dr = 4 pi r - 96.84/r^2
= 0 for max or min of S
4 pi r^3 = 96.84
thnx! :)
how do i find the optimal radius?
We did, read my second answer. I knew that would be next.
4 pi r^3 = 96.84
To develop an equation relating the radius (r) and the surface area (S) of a coffee creamer container, we can use the formulas for the surface area and volume of a cylinder.
The formula for the surface area (S) of a cylinder is:
S = 2πr^2 + 2πrh
The formula for the volume (V) of a cylinder is:
V = πr^2h
Given that the volume of the coffee creamer container is 48.42 cubic inches, we can express the volume equation as:
48.42 = πr^2h
To develop the equation relating the radius (r) and the surface area (S), we need to eliminate the height (h) variable from the equations.
From the volume equation, we can express h in terms of the radius (r) and volume (V):
h = 48.42 / (πr^2)
Now substitute this value of h into the surface area equation:
S = 2πr^2 + 2πr(48.42 / (πr^2))
Simplifying the equation further:
S = 2πr^2 + 96.84 / r
Therefore, the equation relating the radius (r) and the surface area (S) of the coffee creamer container is:
S = 2πr^2 + 96.84 / r