A box with a sqaure base and no top must have a volume of 10

[kimmy_koo51]

A box with a sqaure base and no top must have a volume of 10,000 cm^3. If the smallest dimenson in any direction is 5 cm, then determine the dimensions of the box that minimize that amount of material used.

 

[stapel]

i) Pick a variable for the width of the base of the box. Note that this is also the length of the base.

ii) Pick a variable for the height of the box.

iii) Plug the variables from (i) and (ii) into the volume equation, plugging "10000" in for "V".

iv) Solve (iii) for the height variable in terms of the width variable.

v) Write an expression, in terms of (i), for the area of the base of the box.

vi) Write an expression, in terms of (i) and (ii), for the total surface area of the sides.

vii) Sum (v) and (vi) to get an expression for the total surface area.

viii) Use (iv) to get (vii) in terms only of the variable from (i).

ix) Differentiate.

x) When evaluating max/min points, note that no dimension is allows to be less than "5".

If you get stuck, please reply showing your progress. Thank you.

Eliz.

 

[kimmy_koo51]

This is how far I got:

v= 10,000 cm^3 width is w which is equal to l
v= x^3 height is h
v= 2wh^3
10000 = 2wh^3
17.09/w = h

 

[stapel]

v= x^3

This assumes that the height is the same as the width, and that the box is a cube. You cannot assume this.

Try following the steps provided earlier. Start with picking a variable for the width. For instance, you could use "w". Then the length also is "w". The height could be "h". Then what would the volume be? And so forth.

Eliz.

 

[skeeter]

V = w²h
10000 = w²h
10000/w² = h

surface area, A = w² + 4wh

A = w² + 4w(10000/w²)

find dA/dw and minimize