A rectangular storage container with an open top is to have a volume of 10 m3. The length of this base is twice the width. Material for the base costs $15 per square meter. Material for the sides costs $9 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

Solution:

Given,

Volume = 10 m³

Width = x

Length = 2x

Base area = 2x²

Cost of base = $15

Cost of sides = $9

Since the volume is 10 m³

Volume = base area × height

The height has to be 10/ 2x² 

= 5 /x²

The cost of making such container

Cost of base = 2x² ( 15)

= $ 30x².

Cost of sides = [(2 . 2x . 5 /x²) +(2 . x . 5 /x²)](9)

= $ 270/x.

The overall cost = Cost of base + Cost of sides

f(x)=  30x² + 270/x.

= 30(x² + 9/x)

To get the minimum, let us find the first derivative of f(x) and equate it to zero.

df(x)/dx = 30(2x - 9/x²) = 0

2x - 9/x² = 0

⇒ 2x³ =9

x³ = 4.5

So, x = 1.651 (m)

f(x)= 30x² + 270/x

=30(1.651)² + 270/(1.651)

=81.77 + 163.53

= 245.31 dollars.

Therefore, the cost of materials is 245.31 dollars.

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A rectangular storage container with an open top is to have a volume of 10 m³. The length of this base is twice the width. Material for the base costs $15 per square meter. Material for the sides costs $9 per square meter. Find the cost of materials for the cheapest such container. (Round your answer to the nearest cent.)

Summary:

A rectangular storage container with an open top is to have a volume of 10 m³. The length of this base is twice the width. Material for the base costs $15 per square meter. Material for the sides costs $9 per square meter. The cost of materials for the cheapest such container is 245.31 dollars.