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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: 167 Equation for Volume of this Container: V = ær² (h + ²r) Equation for the Surface Area of this Container: S = 2πrh+ 3r²

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: 167 Equation for Volume of this Container: V = ær² (h + ²r) Equation for the Surface Area of this Container: S = 2πrh+ 3r²

Advanced Engineering Mathematics
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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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
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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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: \[ h = 2.13 \]
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: \[ h = 2.13 \]
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