Gloria would like to construct a box with volume of exactly 45ft^3 using only metal and wood. The metal costs $15/ft^2 and the wood costs $6/ft^2. If the wood is to go on the sides, the metal is to go on the top and bottom, and if the length of the base is to be 3 times the width of the base, find the dimensions of the box (Length, Width, Height) that will minimize the cost of construction. Round your answer to the nearest four decimal places.

Answer 1

L = 3 w

metal area = 2 L w = 6 w^2

wood area = 2 L h + 2 w h

volume = 3 w^2 h = 45 so h =15/w^2

cost = 15*6 w^2 + 6*2 w h
c = 90 w^2 + 12 w h
or
c = 90 w^2 + 12 w (15/w^2)
c = 90 w^2 + 180/w
at max or min dc/dw = 0
0 = 180 w - 180/w^2
1/w^2 = w
w = 1^(1/3) = 1
then L = 3
then h = 15

LOL - lots more cheap wood than expensive metal but check algebra

Answer 2

For this problem, first draw a 3D rectangle and label the width as "x", the length as "3x" and the height as "y". This will help you visualize the area formulas presented.

Area Formulas:
Area of top and base= 2(3x)(x)= 6x^2
*This is multiplied by 2 because we have a top and base*

Area of side= xy
Area of 4 sides=8xy

Volume Formula:
Volume of rectangle=LWH
But in this problem we have L=3W so....

Volume (V)=(3W)(W)(H)= 3HW^2
Now substitute y for H and x for W....

Volume (V)=3yx^2
Desired volume is given in the problem as 45 ft^3.....

45=3yx^2

Now we want to relate the Volume formula with the Total Cost formula and we want to only have one variable to solve for. So, solve for Y using the volume formula we just found....

y=45/(3x^2)

Cost Formulas:
Cost of top and base= $15(6x^2)= 90x^2 dollars
Cost of sides= $6(8xy)= 48xy dollars

Total Cost (C)= 90x^2 + 48xy

Substitute y into the total cost formula:

Total Cost (C)= 90x^2 + 48x(45/(3x^2))
C=90x^2 + 2160x/(3x^2))
C=90x^2 +(720/x)

Now take the derivative of the Total Cost formula to achieve our minimum value:

C'=180x-720x^(-2)
Which can also be written as....
C'=180x-(720/x^2)

Now we want to find the critical numbers of the formula, so get a common denominator of the C' formula and set the numerator =0....
*Note, we don't need to set the denominator =0 because that would be where the function does not exist and we are only interested in values that exist for this function.*

C'= (180x^(3)-720)/(x^2)
0=180x^(3)-720
x^3=4
x=+- 4^(1/3), but we are only concerned with positive values...
x=4^(1/3)
x~1.5874

We are almost done!
Length=3x
Length=3(1.5874)

Width=x
Width=1.5874

Height=y
y=45/(3x^2)
*Plug in 1.5874 for x to get y*

Answer 3

This guy's explanation is weird because he uses L for a length and w for practically the same length. They can both be labeled x's, making the problem less of an eye sore, and h can be labeled y. Your understanding should skyrocket considerably after this redefinition