(2) We have been working on finding minimums and maximums in our calculus class. You need to minimize the surface area of this can to keep cost as low as possible but did you notice that the surface area formula has too many variables? Let's try to simplify this. (a) Use the fact that the manufacturer requires the container's total volume to be kept at 370 cm³ in the volume formula and solve the volume formula for height. Substitute the expression for the height into the surface area formula, so that you have surface area as a function of radius only. Show your work in the space provided.
(2) We have been working on finding minimums and maximums in our calculus class. You need to minimize the surface area of this can to keep cost as low as possible but did you notice that the surface area formula has too many variables? Let's try to simplify this. (a) Use the fact that the manufacturer requires the container's total volume to be kept at 370 cm³ in the volume formula and solve the volume formula for height. Substitute the expression for the height into the surface area formula, so that you have surface area as a function of radius only. Show your work in the space provided.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Help on calculus question (check picture below)
Transcribed Image Text:SOD
(1) Currently cans are in the shape of right circular cylinders. A right circular cylinder diagram is
provided. Look up and write down the formulas for volume and surface area.
Cans of soft drinks are made from aluminum. Due to the soaring prices, the
beverage company executives are looking for all possible cost-cutting measures.
You are being asked to find a way to redesign the traditional can to minimize the
cost. The manufacturer requires the container's total volume to be kept at 370 cm³.
(2) We have been working on finding minimums and maximums in our calculus class. You need to
minimize the surface area of this can to keep cost as low as possible but did you notice that the
surface area formula has too many variables? Let's try to simplify this.
(a) Use the fact that the manufacturer requires the container's total volume to be kept at 370 cm³
in the volume formula and solve the volume formula for height. Substitute the expression for
the height into the surface area formula, so that you have surface area as a function of radius
only. Show your work in the space provided.
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Transcribed Image Text:(b) Use Desmos to graph the surface area function on the following coordinate system. Label
each axis with the variable and measurement units. Identify the point on the graph which is the
minimum. Remember that we are measuring lengths in centimeters.
800
10
an
46-741
Kiad
(c) Refer the graph in Desmos and zoom in on the minimum point. What is the radius at which the
surface area is a minimum? At that radius, what is the surface area of the can? Give both
approximate answers labeled with measurement units.
(3) Now let's use calculus to find that same minimum surface area. In the space provided, show how
to find the derivative of the surface area function. Set the derivative equal to zero to find any
critical point and use the first derivative test to show at what radius the minimum surface area
occurs.
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