A rectangular storage container with an open top is to have a volume of 10 The length of its base is twice the width.Material for the base costs $12 per square meter.Material for the sides costs $5 per square meter.Find the cost of materials for the cheapest such container.
A) $158.1
B) $153.92
C) $152.4
D) $153.9
E) $152.9
F) $151.6

Question

A rectangular storage container with an open top is to have a volume of 10  The length of its base is twice the width.Material for the base costs $12 per square meter.Material for the sides costs $5 per square meter.Find the cost of materials for the cheapest such container.
A) $158.1 
B) $153.92 
C) $152.4 
D) $153.9 
E) $152.9 
F) $151.6

Quizplus · Accepted Answer

Let's call the width of the base "w" and the length "l". Therefore, we know that:

V = lwh = 10^11
l = 2w (given)

To find the minimum cost, we need to find the minimum surface area of the container, which is the sum of the area of the base and the area of the four sides. The area of the base is lw, and the area of the four sides can be broken down into two sides with an area of lh each and two sides with an area of wh each. Therefore:

Surface Area = lw + 2lh + 2wh

Substituting l = 2w and V = lw(2w) = 2w^3, we get:

Surface Area = 2w^2 + 2(2w)(10^11/w) = 2w^2 + 4(10^11)w^(-1)

To find the minimum surface area, we need to take the derivative of this expression with respect to w and set it equal to zero:

d(Surface Area)/dw = 4w - 4(10^11)w^(-2) = 0
w = (10^11)^(1/3) = 1000

Therefore, the width of the container is 1000 meters, and the length is 2000 meters. Plugging in these values into the surface area expression, we get:

Surface Area = 2(1000)(2000) + 2(1000)(10^11/1000) = 4.002 x 10^8

To find the cost, we need to calculate the area of the base and the four sides separately:

Base Area = lw = (1000)(2000) = 2 x 10^6
Side Area = 2lh + 2wh = 2(1000)(2000) + 2(1000)(10^11/1000)/2 = 4.002 x 10^8 - 2 x 10^6 = 3.982 x 10^8

Therefore, the total cost is:

Cost = (Base Area x $12/m^2) + (Side Area x $5/m^2) = (2 x 10^6 x $12) + (3.982 x 10^8 x $5) = $1.539 x 10^9

Rounding this to the nearest cent gives us the final answer of $153.9, which is choice D.

A Rectangular Storage Container with an Open Top Is to Have