Answer:
the dimensions of the box that minimizes the cost are 5 in x 40 in x 40 in
Step-by-step explanation:
since the box has a volume V
V= x*y*z = b=8000 in³
since y=z (square face)
V= x*y² = b=8000 in³
and the cost function is
cost = cost of the square faces * area of square faces + cost of top and bottom * top and bottom areas + cost of the rectangular sides * area of the rectangular sides
C = a* 2*y² + a* 2*x*y + 15*a* 2*x*y = 2*a* y² + 32*a*x*y
to find the optimum we can use Lagrange multipliers , then we have 3 simultaneous equations:
x*y*z = b
Cx - λ*Vx = 0 → 32*a*y - λ*y² = 0 → y*( 32*a-λ*y) = 0 → y=32*a/λ
Cy - λ*Vy = 0 → (4*a*y + 32*a*x) - λ*2*x*y = 0
4*a*32/λ + 32*a*x - λ*2*x*32*a/λ = 0
128*a² /λ + 32*a*x - 64*a*x = 0
32*a*x = 128*a² /λ
x = 4*a/λ
x*y² = b
4*a/λ * (32*a/λ)² = b
(a/λ)³ *4096 = 8000 m³
(a/λ) = ∛ ( 8000 m³/4096 ) = 5/4 in
then
x = 4*a/λ = 4*5/4 in = 5 in
y=32*a/λ = 32*5/4 in = 40 in
then the box has dimensions 5 in x 40 in x 40 in
7/12*6/14
7*6=42
12*14=168
=42/168
=0.25 =1/4
Answer:
Subtract 11/2y from each side of the equation
Step-by-step explanation:
Answer:
The height is 17 ft
Step-by-step explanation:
The surface area of a rectangular prism is
SA = 2(LW +WH + LH)
where L = length, W = width, and H = height
We know SA = 930, L = 12, and W = 9
930 = 2(12*9 +9H + 12H)
Divide each side by 2
930/2 = 2/2(12*9 +9H + 12H)
465 = (12*9 +9H + 12H)
Combine like terms
465 = 84+21H
Subtract 84 from each side
465-84 = 84-84+21H
357 = 21H
Divide by 21
357/21=21H/21
17 = H
The height is 17 ft
