How do you find the dimensions that minimize the amount of cardboard used if a cardboard box without a lid is to have a volume of #8,788 (cm)^3#?

Answer 1
You set #x# as being the sides, and #h# for the height.
The box will have a square bottom. Then the amount of cardboard used will be: For the bottom: #x*x=x^2# For the sides: #x*h*4#(sides)#=4xh#
Total area : #A=x^2+4xh#
The volume of the box= #x*x*h=8788# from which we can conclude that #h=8788/x^2#
Substituting that into the formula for the area #A#, we get:
#A=x^2+4x*(8788/x^2)=x^2+35152/x#
To find the minimum, we have to differentiate and set to #0#
#A'=2x-35152/x^2=0->2x=35152/x^2# multiply by #x^2#
#2x^3=35152->x^3=17576->x=root 3 17576=26# Substitute: #h=8788//26^2=13#
Answer : The sides will be #26cm# and the height will be #13cm#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To minimize the amount of cardboard used for a cardboard box without a lid with a volume of 8,788 (cm)^3, you need to find the dimensions that minimize the surface area of the box.

Let the length, width, and height of the box be represented by ( l ), ( w ), and ( h ) respectively. The volume of the box is given by the formula ( V = lwh ). Since there's no lid, the surface area ( A ) of the box is given by the formula ( A = 2lw + 2lh + wh ).

To find the dimensions that minimize ( A ), you can use calculus by differentiating ( A ) with respect to ( l ), ( w ), and ( h ), setting the derivatives equal to zero, and solving for each dimension.

However, since you are looking for a specific volume (( V = 8,788 ) ( cm^3 )), you can use the given volume to express one of the variables in terms of the other two using the formula ( V = lwh ). Then substitute this expression into the formula for the surface area ( A ). This will give you a function of two variables. You can then find the minimum of this function using calculus.

Solving this mathematically may involve some algebraic manipulation and calculus techniques, such as partial derivatives and critical points. Once you have the critical points, you can determine which point corresponds to the minimum surface area.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7