Simplify the expression:
1 answer:
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Here is the full question
A snack food company wishes to have a cylinder package for it's almond and cashew mix. The cylinder must contain 120 cm³ worth of product. The base and sides will cost $.01 per cm2 to produce but the top, which is plastic and resealable, will cost $.02 per cm2 to produce. What should the dimensions be to minimize cost?
Answer:
The radius and height are both dimension in the cylinder; in order to minimize the cost
radius = 2.515 cm
height = 18.93 cm
Step-by-step explanation:
We denote the radius of the cylinder to be = r
and the height of the cylinder = h
The volume of a cylinder is known to be = πr²h
Also, from the question; we are also told that the cylinder contains 120 πcm³
i.e πr²h = 120π
Dividing both sides with π; we have:
r²h = 120
The base and sides will cost $.01 per cm² to produce
Total cost of the base and side = 0.01 ( πr² + 2πrh)
but the top, which is plastic and resealable, will cost $.02 per cm² to produce.
i.e
cost of the top cylinder = 0.02 ( πr²)
Overall Total cost =
= 0.01 ( πr² + 2πrh) + 0.02 ( πr²)
= 0.01 πr² + 0.02 πrh + 0.02 πr²
=
=
Taking the differentiation to find the radius dimension to minimize cost; we have:
⇒
cm
However,
Therefore; we can say that the cost is minimum at r = 2.515 since it is positive.
To determine the height ; we have:
h = 18.93 cm