SOLUTION: A rectangular piece of cardboard with perimeter 30 in., two parallel and equally spaced creases are made. The cardboard is then folded to make a prism with open ends that are equil

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Question 164928: A rectangular piece of cardboard with perimeter 30 in., two parallel and equally spaced creases are made. The cardboard is then folded to make a prism with open ends that are equilateral triangles. Find the volume of the prism as a function of x and what is the domain of V(x)? Thanks to those who can help!
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A rectangular piece of cardboard with perimeter 30 in., two parallel and equally spaced creases are made. The cardboard is then folded to make a prism with open ends that are equilateral triangles. Find the volume of the prism as a function of x and what is the domain of V(x)?
:
Let x = the 3 equal distances, between the two folds
then
3x = width of the cardboard
and
Let L = the length of cardboard
;
Perimeter:
2(3x) + 2L = 30
Simplify divide by 2
3x + L = 15
L = (15-3x)
:
Find the height (h) of the equilateral triangle with sides = x
h^2 + (x/2)^2 = x^2
h^2 = x^2 - (x/2)^2
h = sqrt%28x%5E2-%28x%2F2%29%5E2%29
:
Vol of a triangular prism = 1%2F2*L*w*h
In this problems
L = (15-3x)
w = x
h = sqrt%28x%5E2-%28x%2F2%29%5E2%29
:
V(x) = 1%2F2*(15-3x)*x*sqrt%28x%5E2-%28x%2F2%29%5E2%29
:
You can tell by L, (15-3x), that x has to be less than 5 (the domain)
:
If you graph this:

The max volume when x = 3.33, Domain 0 to <+5