Question

"rectangular storage container with a top made of the same materials as its sides is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $10 per square meter. Material for the sides and top costs $6 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)"

Answer 1

Answer:

Cost = $0.08

Step-by-step explanation:

Volume of the Container =$10m^3$

Volume of a Cuboid =lwh

Since the length of the base is twice the width: l=2w

$2w*h*w=10\\2w^{2}h=10$

Total Surface Area of a Cuboid = 2(LW+LH+WH)

Material for the base costs $10 per square meter.

Cost of the Base = $10 X LW = $(10LW)

Cost of the Sides and the top =$6(LW+2(LH+WH))=$(6LW+12LH+12WH)

Total Cost=$(10LW)+$(6LW+12LH+12WH)

              =16LW+12LH+12WH

From the volume,

$\\2w^{2}h=10\\h=\frac{10}{2w^{2}} =5w^{2}$

Also, l=2w

Substituting H and L into 16LW+12LH+12WH

$16(2W)W+12(2W)(5W^2)+12W(5W^2)\\Cost(W)=32W^2+120W^3+10W^3\\Cost(W)=32W^2+240W^3$

The Minimum Value of C(W) is at the point where the derivative=0.

$C^{'}(W)=64W+720W^2\\If C^{'}(W)=0\\64W+720W^2=0\\720W^2=-64W\\720W=-64\\W=-\frac{4}{45}$

$Cost(W)=32(-\frac{4}{45})^2+240(-\frac{4}{45})^3\\Cost = 0.08$

Answer 2

Answer:

$165.75

Step-by-step explanation:

Let's assume the width to be = x

Since the length is twice that of the base,then the lenght will be = 2x

Let height = y

with the above variables, we set an equation for volume and cost.

10 = 2x × y = volume

Volume: V = Lwh

10 = (2x)(x)(y)

10 = 2yx²

y = 5/x²

Cost: C(w) = 10(Lw) + 2[6(hw)] + 2[6(hL)])

= 10(2x²) + 2(6(yx)) + 2(6(y)(2x)

= 20x² + 2[6w(5/x²)] + 2[12w(5/x²)]

= 20w^2 + 60/w + 120/w

= 20x² + 180x^(-1)

C'(x) = 40x - 180x^(-2)

Critical numbers:

(40x³- 180)/x² = 0

40x³-180 = 0

40x³= 180

x³= 9/2

x = 1.65 m ( the width)

2x= 3.30 m (the length)

y = 1.84 m (the height)

Cost: C = 10(Lw) + 2[6(hw)] + 2[6(hL)])

= 10(3.30)(1.65) + 2[6(1.84)(1.65)] + 2[6(1.84)(3.30)])

= $165.75 cheapest cost

The cost of materials will be cheapest when the width of the

container is 1.65 meters, base length is 3.30 meters, and the height is

1.84 meters.