calc help plz?

Question

katie is building a wooden rectangular toy storage for her younger brother. The box will have an open top and volume of 9m^3. For design purposes, Katie would like the length of its base to be triple its width. Thick wood for the base costs $8/m^2 and wood for the sides costs $5/m^2.

Express the cost of wood as a function of the width of the base.

What is a suitable doman and range for the function in this context?

Answer 1

L = 3 w
vol = w (3w) h = 9
h = 3 / w^2

cost = $8 (3w)(w) + $5 * 2 (3 / w^2 ) [ 3w + w ]
cost = $24 w^2 + $24 / w

h must not be less or equal to zero

Answer 2

The area of the bottom is

A = L*b but L=3*b
A=3b^2

the height is given by the volume = 9

V = A*h = 3b^2*h => h = 3/b^2

Cost = $8/m^2*A + $5/m^2*2*(3b+b)*3/b^2
Cost = 24b^2 + 120/b $

a suitable domain is b>0 becuase a negative length is meaninless. a suitable range should also be cost>0.

Answer 3

the length of its base to be triple its width.
means

L = 3 * W

Area of the rectangle Base = L * W = (3W) * W = 3W^2
Cost (base) = 8 * (3W^2) = 24 W^2

Volume (box) = base area * height
so, height = volume /base area = 9 / (3W^2) = 3 /W^2

Area of 4 Sides of the box = perimeter of the base * height
area = 2 (L + W) * Height
= 2 * (3W + W) * (3/W^2)
= 2 * (4W) * 3/W^2
= 24/W

cost of 4 sides =5 * (24/W) = 120/W

total cost = cost (base) = cost (sides)
= 24W^2 + 24/W
= 24 * (W^2+ 1/W)

Answer 4

since L = 3w, area of base B = 3w².
perimenter of base P = 2(L+w) = 8w
height = volume / B = 9/(3w²) = 3/w².
area of sides A = Ph = 8w(3/w²) = 24/w

cost = 8B + 5A = 8(3w²) + 5(24/w)
cost = 24w² + 120/w

as w→0, B→0 and all the volume is in the height, h→∞.
as w increases, B increases and h→0, all the volume is in B.

as for domain and range, w > 0 is obvious for domain. function minimizes when
C' = 0 = 48w - 120/w²
120/w² = 48w
2.5 = w^3
w = 2.5^(1/3) = 1.357 m, so range is [1.357,∞)

adding context, w in (0.5,3) and c in (135,200) makes sense.

Answer 5

Using Lwh=9 and L=3w, with L being length, w being width, and h being height, then 8wL+5*2wh+5*2Lh =
24w^2+30/w+90/w =24w^2+120/w.

So, cost = 24w^2+120/w.