Answer:
Base = Length = (2V/5)^⅓
Height = V^⅓/(2/5)^⅔
Step-by-step explanation:
Let the sides of the aquarium be l,b and h
Where l = length
b = base
h = height
Volume = lbh
V = lbh
The surface area of the box is
A = 2bh + 2lh + 2lb
We'll replace the 2lb with 5lb to get the cost of area because it's given the question that the slate (base) of the box cost 5 times as much per unit area as glass
C(l,b,h) = 2bh + 2lh + 5lb
Make h the subject of formula in (V = lbh)
h = V/lb
This will enable us to account for the lowest cost of materials by taking derivatives of the cost equation, and we need to solve for a l to put into the solution.
Substitute V/lb for h in the cost equation
C(l,b,V/lb) = 2b(V/lb) + 2l(V/lb) + 5lb ------ Simplify
C(l,b,V/lb) = 2V/l + 2V/b + 5lb
Take the derivatives of the above with respect to l and b
Cl = -2V/l² + 5b
Cb = -2V/b² + 5l
Equate Cb to Cl (this implies that b = l)
Cb = Cl =>
-2V/l² + 5b = -2V/b² + 5l
So, we have
C(b,b) = -2V/b² + 5b = 0
-2V/b² + 5b = 0 ---- Solve for b
-2V/b² = -5b ---- Multiply through by b²
-2V = -5b³ ---- Divide through by -5
2V/5 = b³ ---; Rearrange
b³ = 2V/5
b = (2V/5)^⅓
b = l
So, l = (2V/5)^⅓
h = V/lb ---- (b = l)
h = V/l² ---- Substitute (2V/5)^⅓ for l
h = V/((2V/5)^⅓)²
h = V/(2V/5)^⅔
h = V/((V^⅔)(2/5)^⅔)
h = V^⅓/(2/5)^⅔
