Two Very Difficult Optimization Problems! Help!

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April 8th 2008, 10:19 PM
Hibijibi

Two Very Difficult Optimization Problems! Help!

Here are two problems that have me scratching my head, any and all help is highly appreciated. And I'll love you forever.

1. A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross-section) of 100 inches. Use calculus to find the dimensions of the package of a maximum volume that can be sent. (Note: in this case each cross-section is a circle.

2. You are driving home from the grocery store and your car runs out of gas. You want to get to the gas station as quickly as possible so that your groceries don't spoil. You are located as show in the figure. You can walk 4MPH on the road but only 3MPH through the field. Find the value of 0 (Theta) that will minimize the time it takes you to reach the gas station. Give a formula for the function you are trying to minimize. Use calculus to find the minimum point.) Explain why the closed interval method is appropriate. Also explain how you know the function achieves an absolute minimum at the value you claim.

Here's a rough sketch using the keyboard of the figure
______________________
| ...._________________ Gas station
| ....|<------7miles----->
| ....| ......./
|5mi|...... /
| ....| ..../ Field
| ....|... /
| ....| ../
| ....|0/
Car

0 Represents theta, and yes that is a triangle. Field and road shown, 5 miles up, and 7 across. All help is appreciated beyond what you know.
April 8th 2008, 10:49 PM
CaptainBlack

Quote:

Originally Posted by Hibijibi

Here are two problems that have me scratching my head, any and all help is highly appreciated. And I'll love you forever.

1. A cylindrical package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross-section) of 100 inches. Use calculus to find the dimensions of the package of a maximum volume that can be sent. (Note: in this case each cross-section is a circle.

Since the package is cylindrical let $l$ be the length and $r$ be the radius in inches.

Then for a maximal package the constraint is tight so we have:

$l+2 \pi r=100\ \ \ \ ...(1)$

The volume is:

$V=\pi r^2 l\ \ \ \ \ \ ...(2)$

Now substitute $l$ from (1) into (2). Now you have an unconstrained optimisation problem.

RonL
April 8th 2008, 10:53 PM
CaptainBlack

Quote:

Originally Posted by Hibijibi

2. You are driving home from the grocery store and your car runs out of gas. You want to get to the gas station as quickly as possible so that your groceries don't spoil. You are located as show in the figure. You can walk 4MPH on the road but only 3MPH through the field. Find the value of 0 (Theta) that will minimize the time it takes you to reach the gas station. Give a formula for the function you are trying to minimize. Use calculus to find the minimum point.) Explain why the closed interval method is appropriate. Also explain how you know the function achieves an absolute minimum at the value you claim.

Here's a rough sketch using the keyboard of the figure
______________________
| ...._________________ Gas station
| ....|<------7miles----->
| ....| ......./
|5mi|...... /
| ....| ..../ Field
| ....|... /
| ....| ../
| ....|0/
Car

0 Represents theta, and yes that is a triangle. Field and road shown, 5 miles up, and 7 across. All help is appreciated beyond what you know.

See earboth's solution to question 2 in this thread:http://www.mathhelpforum.com/math-he...tml#post126210

RonL