Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Transcribed Image Text:Constructing the Fuel Tank
"Great, I've got the fuel, but how do I get it to a spacecraft? Computer, how do I transport this fuel?" Your
computer responds that the burning of the fuel destroys the container that it is held in, and that a new container
is constructed for each launch of the spacecraft. The shape of the container is a cylinder with a hemisphere
attached to the bottom. Your computer is able to generate an image (not to scale) and equations for the volume
and surface area of this shape.
You see a warning notice that supplies of materials for constructing this container are low. In order to create the
container with limited supplies, you will need to provide the height and radius (to 2 decimal places) that
minimizes the surface area of the container. What values do you provide?
Total Volume of Fuel: 16T
Equation for Volume of this Container: V = πr² (h+
Equation for the Surface Area of this Container: S = 2πrh+ 3πr²
3-
0 0.5 11.5 2
Radius: Number
Height: Number
1.5 10.50
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Follow-up Question
how do you get from start to finsh in the pic pleaseexplain in detail
![To minimize the surface area and differentiate with respect to \( r \):
\[
\frac{d}{dr} S = \frac{d}{dr} \left( \pi \left( \frac{32}{r} + \frac{5}{3} r^2 \right) \right)
\]
\[
\Rightarrow \frac{d}{dr} S = \pi \left( -\frac{32}{r^2} + \frac{10}{3} r \right)
\]
Now, set this value to zero and solve for \( r \):
\[
r = 2.13
\]
Substitute the value of \( r \) back into Step 1 to find:
\[
h = 2.13
\]](https://content.bartleby.com/qna-images/question/f0c2c17c-c864-4c22-95dc-8933004abe78/26086fc6-deba-4040-a0d7-b4f650aaaa8c/714igu_processed.png)
Transcribed Image Text:To minimize the surface area and differentiate with respect to \( r \):
\[
\frac{d}{dr} S = \frac{d}{dr} \left( \pi \left( \frac{32}{r} + \frac{5}{3} r^2 \right) \right)
\]
\[
\Rightarrow \frac{d}{dr} S = \pi \left( -\frac{32}{r^2} + \frac{10}{3} r \right)
\]
Now, set this value to zero and solve for \( r \):
\[
r = 2.13
\]
Substitute the value of \( r \) back into Step 1 to find:
\[
h = 2.13
\]
Solution
by Bartleby Expert
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
how do you get from start to finsh in the pic pleaseexplain in detail
Transcribed Image Text:To minimize the surface area and differentiate with respect to \( r \):
\[
\frac{d}{dr} S = \frac{d}{dr} \left( \pi \left( \frac{32}{r} + \frac{5}{3} r^2 \right) \right)
\]
\[
\Rightarrow \frac{d}{dr} S = \pi \left( -\frac{32}{r^2} + \frac{10}{3} r \right)
\]
Now, set this value to zero and solve for \( r \):
\[
r = 2.13
\]
Substitute the value of \( r \) back into Step 1 to find:
\[
h = 2.13
\]
Solution
by Bartleby Expert
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