Question

question 4
a microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 cubic centimeters of soup. the sides and bottom of the container will be made of styrofoam costing 0.03 cents per square centimeter. the top will be made of glued paper, costing 0.08 cents per square centimeter. find the dimensions for the package that will minimize production cost.

helpful information:
h : height of cylinder, r : radius of cylinder

volume of a cylinder: ( v = pi r^2 h )

area of the sides: ( a = 2pi r h )

area of the top/bottom: ( a = pi r^2 )

to minimize the cost of the package:
radius: (\boxed{quad}) cm
height: (\boxed{quad}) cm
minimum cost: (\boxed{quad}) cents

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Explanation:

Step1: Express \(h\) in terms of \(r\)

Given \(V=\pi r^{2}h = 350\), so \(h=\frac{350}{\pi r^{2}}\)

Step2: Write the cost function \(C(r)\)

The cost of the bottom is \(0.03\pi r^{2}\), the cost of the sides is \(0.03\times2\pi rh\), and the cost of the top is \(0.08\pi r^{2}\).

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Step3: Find the derivative of \(C(r)\)

$$ C^{\prime}(r)=0.22\pi r-\frac{21}{r^{2}} $$

Set \(C^{\prime}(r) = 0\)

$$ 0.22\pi r-\frac{21}{r^{2}}=0 $$

$$ 0.22\pi r^{3}=21 $$

$$ r^{3}=\frac{21}{0.22\pi}\approx\frac{21}{0.6912}\approx30.38 $$

$$ r\approx3.12 $$

Step4: Find \(h\)

Substitute \(r\approx3.12\) into \(h = \frac{350}{\pi r^{2}}\)

$$ h=\frac{350}{\pi\times(3.12)^{2}}\approx\frac{350}{30.57}\approx11.45 $$

Step5: Find the minimum cost

Substitute \(r = 3.12\) into \(C(r)\)

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Answer:

Radius: \(3.12\) cm
Height: \(11.45\) cm
Minimum cost: \(10.1\) cents