Extreme Value: closed box w/ sq. base; ships' closest dist.

[Ahe]

Ok I have two homework problems that I'm almost positive are dealing with extreme values. I've usually done those pretty easily with graph functions, but these are word problems...They're really confusing me and I'm not even sure how to start. Help?

1: A closed rectangular container with a square base is to have a volume of 2250 inches cubed. The material for the top and bottom of the container will cost $2 per inch squared and the material for the sides will cost $3 per inch squared. Find the dimensions of the container of the least cost.

2: At noon ship A was 100 miles due east of ship B. Ship A is sailing west at 12 miles per hour and ship B is sailing south at 10 miles per hour. At what time will the ships be nearest to each other, and what will this distance be?

Thanks!

 

[Deleted member 4993]

Re: Extreme Value Word Problems

Ok I have two homework problems that I'm almost positive are dealing with extreme values. I've usually done those pretty easily with graph functions, but these are word problems...They're really confusing me and I'm not even sure how to start. Help?

1: A closed rectangular container with a square base is to have a volume of 2250 inches cubed. The material for the top and bottom of the container will cost $2 per inch squared and the material for the sides will cost $3 per inch squared. Find the dimensions of the container of the least cost.

2: At noon ship A was 100 miles due east of ship B. Ship A is sailing west at 12 miles per hour and ship B is sailing south at 10 miles per hour. At what time will the ships be nearest to each other, and what will this distance be?

Thanks!

1)

What is the problem asking you to do - Find the dimensions of the container....

Well then what are the dimensions - Length - Width - Height

So those are our variables -

Let

Length = L

Width = W

Height = H

There is another part of the question - minimum cost ....

What causes the cost - the cost of the material to make the sidewalls and the top/bottom

How do we figure out the material needed to make the sides (and the top/bottom)?

Since material cost is in sq.ft(area) - how much area of the material I need to buy?

What are the areas of the sidewalls (and the top/bottom) - in terms of L, W & H?

That's a start - can you continue?

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[galactus]

Re: Extreme Value Word Problems

2: At noon ship A was 100 miles due east of ship B. Ship A is sailing west at 12 miles per hour and ship B is sailing south at 10 miles per hour. At what time will the ships be nearest to each other, and what will this distance be?

This is an exercise in ol' Pythagoras. You know, many of these optimization problems are very cliche. If you google, you will most likely find something similar.

Anyway, make a diagram and label the ships A and B.

The distance A is from where B started is 100-12t.

Since B is sailing south. it's distance from where it started is just 10t.

This forms a right triangle. Now, set up the function to differentiate, differentiate it, set to 0 and solve for t.

Then, add that time to 12 noon. That's it.

Remember, a handy thing to do when max or minning a distance is to realize that the distance and the square of the distance have their max and min at the same point. In other words, you do not need the radical to get the correct solution. It makes the computations easier.