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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.)
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.
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.
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