The "Mean" Value Theorem

Willy the Wallaby is back and this time he is not looking for a job! This time he needs to build a cage fo0r his new pet Bob, a boa constrictor. The volume of the cage has to be 4000 ft³ and the length is 5 x the height of the cage. That is not all Wally has to pay with his own money, probably from his new job, to build the cage. So Wally wants to build the least expensive cage that has the correct volume, length and height. For the materials it cost $ .25/sq ft

Even though I worded it differently it still has the information for what you need.

1) What are the dimensions of the least expensive cage?
And Mr. benofschool was kindly enough to show us how it was done. To start things off first list down the information that is given to you
height = h
length = 5h
volume = 4000 ft³
width = w

Seeing that the width and height are unknown and that the length is determined by the height, we find the width first.
Then we find the surface area of the cage, which is needed to solve for the total amount it will cost Wally to build it.

At this point you find the derivative of the equation

Next you let "h" go to zero to find what "h" is at that point.

and prove that "h" is what it is by using the first derivative test

and finally using the new found height you can now solve for the rest of the dimensions of the cage.

2)How much does it cost?
For this question there is nothing special about it just fill in the numbers \:

After taking most of the class time to solve the one question we move on to the beginning of the "mean" value
In this graph the first red line (i)is called a secant line, which is the mean, or the average of the line f(x).

If this was a graph of a person's change in velocity while driving, he would have his license suspended, but since this is in a Calculus class we don't even care if that person was fined or not. In the changing velocity graph the secant line is the person's average driving velocity, and the only once that the person drove at his/her said average, which is where the secant line cross the f(x) line.

Plus another thing about this graph is that at point c (both of them) the slope at those points are the same as the slope of the secant line, but to find the slope of a secant line, you need to use this formula:


Finally the next scriber will be Yinan,
*rants*
and because a certain someone had me to scribe today I had to cut my hours short from volunteering at the hospital helping the community and making people happy to get to feel better in this crummy weather!!! Which had added a lot of work on me for today!! And now I will have to stay up late to make up for my lost hours typing this post!! And as for Mr. Racoon this is what he thinks of you when you get me post on a Tuesday night.